*********************** q = 3 x = 3 No solutions of the modular equation No feasible patterns *********************** q = 3 x = 4 No solutions of the modular equation No feasible patterns *********************** q = 3 x = 5 Group 1 Possible row sums: [2, 6, 10, ] Possible col sums: [3, 7, 11, ] Processing group 1 PG(3,3), x = 5, u in L: [0, 0, 0, 1, ] [2, 2, 2, 3, ] [2, 2, 2, 3, ] [2, 2, 2, 3, ] [0, 1, 2, 2, ] [0, 1, 2, 2, ] [1, 2, 3, 3, ] [1, 2, 3, 3, ] PG(3,3), x = 5, u not in L: [0, 0, 0, 2, ] [1, 1, 1, 3, ] [1, 1, 1, 3, ] [1, 1, 1, 3, ] [0, 0, 1, 1, ] [0, 0, 1, 1, ] [1, 1, 2, 2, ] [2, 2, 3, 3, ] #{Class patterns} = 2 #{Non-class patterns} = 2 RowSums: [2, 6, 10, ] ColSums: [3, 7, 11, ] Sle with plane and point types equations 0 1 0 0 = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 1 0 0 = 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 2 0 0 = 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 3 2 = 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 2 = 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 = 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 6 2 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 4 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 3 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 6 0 0 2 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 2 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 | 0 0 2 0 0 = 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 | 0 3 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 | 0 0 0 1 2 = 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 | 0 0 0 3 1 = 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 | 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 | 0 0 0 0 2 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 3 2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 3 0 0 2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 2 6 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 4 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 6 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 1 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 65 | 0 0 0 1 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 65 | 0 9 0 0 -4 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 -2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 6 -9 -2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 4 0 -4 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 4 -9 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 4 0 -4 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 9 2 0 -6 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 Row-reduced echelon form: 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 | 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 | 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 | 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 | 0 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -10 | 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -225 | 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -90 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -10 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -90 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -225 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -210 | 0 The number of equations: 22 Rank: 4 The number of unknowns z_i: 4. The number of indeterminates: 19 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 = 40 3 7 11 = 260 3 21 55 = 910 Top bound on feasible weight distribution on planes (without Zero equations): 15 15 10 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 = 15 0 1 0 = 15 0 0 1 = 10 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 = 15 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 10 0 0 0 1 0 0 0 0 0 = 210 0 0 0 0 1 0 0 0 0 = 225 0 0 0 0 0 1 0 0 0 = 150 0 0 0 0 0 0 1 0 0 = 210 0 0 0 0 0 0 0 1 0 = 150 0 0 0 0 0 0 0 0 1 = 90 1 0 0 0 0 0 0 0 0 = 15 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 10 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 = 15 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 10 0 0 0 1 0 0 0 0 0 = 210 0 0 0 0 1 0 0 0 0 = 225 0 0 0 0 0 1 0 0 0 = 150 0 0 0 0 0 0 1 0 0 = 210 0 0 0 0 0 0 0 1 0 = 150 0 0 0 0 0 0 0 0 1 = 90 Top bound on feasible weight distribution on planes: 15 15 10 Feasible weight distribution of planes 1: [15, 15, 10, ] The number of solutions of the equations from Lemma 3.1: 1 The number of solutions of the extended system:1 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 = 40 2 6 10 = 260 1 15 45 = 910 Top bound on feasible weight distribution on points (without Zero equations): 10 15 15 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 = 10 0 1 0 = 15 0 0 1 = 15 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 = 10 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 15 0 0 0 1 0 0 0 0 0 = 90 0 0 0 0 1 0 0 0 0 = 150 0 0 0 0 0 1 0 0 0 = 150 0 0 0 0 0 0 1 0 0 = 210 0 0 0 0 0 0 0 1 0 = 225 0 0 0 0 0 0 0 0 1 = 210 1 0 0 0 0 0 0 0 0 = 10 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 15 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 = 10 0 1 0 0 0 0 0 0 0 = 15 0 0 1 0 0 0 0 0 0 = 15 0 0 0 1 0 0 0 0 0 = 90 0 0 0 0 1 0 0 0 0 = 150 0 0 0 0 0 1 0 0 0 = 150 0 0 0 0 0 0 1 0 0 = 210 0 0 0 0 0 0 0 1 0 = 225 0 0 0 0 0 0 0 0 1 = 210 Top bound on feasible weight distribution on points: 10 15 15 Feasible weight distribution of points 1: [10, 15, 15, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 1: [15, 15, 10, ] Feasible weight distribution of points 1: [10, 15, 15, ] 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 | 20 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 | 45 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 | 20 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 | 45 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -10 | 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -225 | 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -90 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -10 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -15 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -90 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -150 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -210 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -225 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -210 | 0 Top bound on solutions: 20 45 20 45 Solutions count: 1 Solutions: [20, 45, 20, 45, ] Total solutions count: 1 *********************** q = 4 x = 3 No solutions of the modular equation No feasible patterns *********************** q = 4 x = 4 No solutions of the modular equation No feasible patterns *********************** q = 4 x = 5 Group 1 Possible row sums: [0, 5, 10, 15, 20, ] Possible col sums: [0, 5, 10, 15, 20, ] Processing group 1 PG(3,4), x = 5, u in L: [0, 0, 0, 2, 2, ] [0, 0, 0, 2, 2, ] [0, 0, 0, 2, 2, ] [2, 2, 2, 4, 4, ] [2, 2, 2, 4, 4, ] PG(3,4), x = 5, u not in L: [0, 0, 0, 0, 0, ] [0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] #{Class patterns} = 1 #{Non-class patterns} = 2 RowSums: [0, 5, 10, 15, ] ColSums: [0, 5, 10, 15, ] Sle with plane and point types equations 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 0 0 = 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 = 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 5 1 = 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 4 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 6 20 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 6 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 5 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 4 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 6 0 20 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 6 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 105 | 0 0 1 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 252 | 0 36 -5 -5 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 -20 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 12 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 -20 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 12 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 Row-reduced echelon form: 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 32 Rank: 3 The number of unknowns z_i: 3. The number of indeterminates: 29 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 = 85 0 5 10 15 = 525 0 10 45 105 = 2100 Top bound on feasible weight distribution on planes (without Zero equations): 22 85 42 14 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 = 22 0 1 0 -3 = 21 0 0 1 3 = 42 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 = 22 0 1 0 -3 0 0 0 0 0 0 0 0 0 0 = 21 0 0 1 3 0 0 0 0 0 0 0 0 0 0 = 42 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 22 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 = 21 0 0 1 3 0 0 0 0 0 0 0 0 0 0 = 42 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 = 85 0 5 10 15 = 525 0 10 45 105 = 2100 Top bound on feasible weight distribution on points (without Zero equations): 22 85 42 14 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 = 22 0 1 0 -3 = 21 0 0 1 3 = 42 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 = 22 0 1 0 -3 0 0 0 0 0 0 0 0 0 0 = 21 0 0 1 3 0 0 0 0 0 0 0 0 0 0 = 42 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 22 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 = 21 0 0 1 3 0 0 0 0 0 0 0 0 0 0 = 42 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 4 x = 6 Group 1 Possible row sums: [0, 5, 10, 15, 20, ] Possible col sums: [1, 6, 11, 16, 21, ] Processing group 1 PG(3,4), x = 6, u in L: [0, 0, 0, 2, 2, ] [0, 0, 0, 2, 2, ] [1, 1, 1, 3, 3, ] [2, 2, 2, 4, 4, ] [2, 2, 2, 4, 4, ] [0, 1, 1, 1, 1, ] [0, 1, 1, 1, 1, ] [0, 1, 1, 1, 1, ] [2, 3, 3, 3, 3, ] [3, 4, 4, 4, 4, ] [0, 1, 2, 2, 4, ] [0, 1, 2, 2, 4, ] [0, 1, 2, 2, 4, ] [0, 1, 2, 2, 4, ] [0, 1, 2, 2, 4, ] PG(3,4), x = 6, u not in L: [0, 0, 0, 0, 0, ] [0, 0, 0, 0, 0, ] [2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, ] [0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, ] [3, 3, 3, 3, 3, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [1, 1, 2, 3, 3, ] #{Class patterns} = 3 #{Non-class patterns} = 3 RowSums: [0, 5, 10, 15, 20, ] ColSums: [1, 6, 11, 16, 21, ] Sle with plane and point types equations 0 0 1 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 1 1 0 0 0 = 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 4 2 0 0 0 = 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 0 0 0 = 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 0 0 = 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 2 = 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 5 5 1 = 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 2 = 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 | 0 2 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 12 2 0 0 -15 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 4 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 Row-reduced echelon form: 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 47 Rank: 6 The number of unknowns z_i: 6. The number of indeterminates: 41 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 85 1 6 11 16 21 = 630 0 15 55 120 210 = 2835 Top bound on feasible weight distribution on planes (without Zero equations): 24 85 48 16 8 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 24 0 1 0 -3 -8 = 13 0 0 1 3 6 = 48 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 24 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 13 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 48 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 24 0 0 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 13 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 48 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 = 85 0 5 10 15 20 = 630 0 10 45 105 190 = 2835 Top bound on feasible weight distribution on points (without Zero equations): 22 85 63 21 7 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 22 0 1 0 -3 -8 = 0 0 0 1 3 6 = 63 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 63 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 63 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 4 x = 7 Group 1 Possible row sums: [1, 6, 11, 16, 21, ] Possible col sums: [1, 6, 11, 16, 21, ] Processing group 1 PG(3,4), x = 7, u in L: [0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, ] [3, 3, 3, 3, 3, ] [3, 3, 3, 3, 3, ] [3, 3, 3, 3, 3, ] [0, 0, 0, 0, 0, ] [2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, ] [4, 4, 4, 4, 4, ] [0, 0, 1, 2, 2, ] [0, 0, 1, 2, 2, ] [1, 1, 2, 3, 3, ] [2, 2, 3, 4, 4, ] [2, 2, 3, 4, 4, ] [0, 1, 3, 3, 3, ] [0, 1, 3, 3, 3, ] [0, 1, 3, 3, 3, ] [0, 1, 3, 3, 3, ] [0, 1, 3, 3, 3, ] [0, 2, 2, 2, 4, ] [0, 2, 2, 2, 4, ] [0, 2, 2, 2, 4, ] [0, 2, 2, 2, 4, ] [0, 2, 2, 2, 4, ] [1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, ] [3, 3, 3, 3, 3, ] [4, 4, 4, 4, 4, ] [1, 1, 1, 3, 4, ] [1, 1, 1, 3, 4, ] [1, 1, 1, 3, 4, ] [1, 1, 1, 3, 4, ] [1, 1, 1, 3, 4, ] PG(3,4), x = 7, u not in L: [0, 0, 0, 0, 1, ] [0, 0, 0, 0, 1, ] [2, 2, 2, 2, 3, ] [2, 2, 2, 2, 3, ] [2, 2, 2, 2, 3, ] [0, 0, 0, 0, 1, ] [1, 1, 1, 1, 2, ] [1, 1, 1, 1, 2, ] [1, 1, 1, 1, 2, ] [3, 3, 3, 3, 4, ] [0, 0, 2, 2, 2, ] [0, 0, 2, 2, 2, ] [0, 0, 2, 2, 2, ] [0, 0, 2, 2, 2, ] [1, 1, 3, 3, 3, ] [0, 1, 1, 1, 3, ] [0, 1, 1, 1, 3, ] [0, 1, 1, 1, 3, ] [0, 1, 1, 1, 3, ] [1, 2, 2, 2, 4, ] #{Class patterns} = 7 #{Non-class patterns} = 4 RowSums: [1, 6, 11, 16, 21, ] ColSums: [1, 6, 11, 16, 21, ] Sle with plane and point types equations 0 0 0 1 1 0 0 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 1 0 0 3 0 0 0 0 = 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 5 5 1 0 3 5 0 0 0 0 0 = 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 3 0 0 1 0 0 0 0 = 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 1 0 1 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 -3213 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 -306 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 -21420 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 -44982 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 -199206 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 The number of equations: 49 Rank: 11 The number of unknowns z_i: 11. The number of indeterminates: 41 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 85 1 6 11 16 21 = 735 0 15 55 120 210 = 3675 Top bound on feasible weight distribution on planes (without Zero equations): 24 85 66 23 8 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 24 0 1 0 -3 -8 = -8 0 0 1 3 6 = 69 The extended system (The equations from Lemma 3.1 and the Zero equations): 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 = 2142 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 = 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 = 3213 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 = 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 = 306 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 = 21420 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 = 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 = 44982 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 = 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 = 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 = 199206 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 24 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -8 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 69 Row-reduced echelon form of the extended system: 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 = 2142 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 = 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 = 3213 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 = 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 = 306 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 = 21420 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 = 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 = 44982 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 = 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 = 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 = 199206 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on planes: 21 53 63 18 1 Feasible weight distribution of planes 1: [6, 46, 15, 18, 0, ] The number of solutions of the equations from Lemma 3.1: 35 The number of solutions of the extended system:1 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 = 85 1 6 11 16 21 = 735 0 15 55 120 210 = 3675 Top bound on feasible weight distribution on points (without Zero equations): 24 85 66 23 8 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 24 0 1 0 -3 -8 = -8 0 0 1 3 6 = 69 The extended system (The equations from Lemma 3.1 and the Zero equations): 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 = 2142 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 = 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 = 3213 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 = 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 = 306 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 = 21420 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 = 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 = 44982 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 = 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 = 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 = 199206 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 24 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -8 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 69 Row-reduced echelon form of the extended system: 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 = 2142 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 = 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 = 3213 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 = 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 = 306 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 = 21420 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 = 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 = 44982 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 = 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 = 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 = 199206 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 = 2142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on points: 21 53 63 18 1 Feasible weight distribution of points 1: [6, 46, 15, 18, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 1: [6, 46, 15, 18, 0, ] Feasible weight distribution of points 1: [6, 46, 15, 18, 0, ] 51 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 0 | 306 0 102 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -7 3 0 2142 | 0 0 0 34 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 30 0 0 | 4590 0 0 0 51 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 -2142 | 306 0 0 0 0 102 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -14 -45 0 4284 | 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 17 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -7 -31 0 2142 | 0 0 0 0 0 0 0 0 102 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -65 -300 0 21420 | 1530 0 0 0 0 0 0 0 0 17 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 10 0 0 | 1530 0 0 0 0 0 0 0 0 0 102 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 180 0 0 | 1530 0 0 0 0 0 0 0 0 0 0 17 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 10 0 0 | 1530 0 0 0 0 0 0 0 0 0 0 0 = 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -21 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 78 0 -8568 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 -1071 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 -2448 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 -180 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -53 -123 0 2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 -315 0 -4284 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -13 -43 0 2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 45 0 -4284 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 67 576 0 -55692 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 390 0 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0 0 0 0 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 27 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -23 -114 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 33 0 -3213 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 306 0 0 0 0 0 0 0 0 0 0 0 0 1 -15 0 -306 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 0 65 300 0 -21420 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 0 0 0 -46 -75 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 137 597 0 -44982 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 -6 5 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 102 0 0 0 0 0 0 0 7 -3 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 -115 -672 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 0 0 0 -115 -570 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 -46 -126 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -3 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 0 0 616 2541 0 -199206 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 -15 -30 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 7 -3 0 -2142 | 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 6 0 135 6 0 0 0 15 90 15 90 Solutions count: 1 Solutions: [6, 0, 135, 6, 0, 0, 0, 15, 90, 15, 90, ] Total solutions count: 1 *********************** q = 4 x = 8 No solutions of the modular equation No feasible patterns *********************** q = 5 x = 3 No solutions of the modular equation No feasible patterns *********************** q = 5 x = 4 Group 1 Possible row sums: [0, 6, 12, 18, 24, 30, ] Possible col sums: [4, 10, 16, 22, 28, ] No feasible patterns for class No feasible patterns *********************** q = 5 x = 5 Group 1 Possible row sums: [1, 7, 13, 19, 25, 31, ] Possible col sums: [4, 10, 16, 22, 28, ] Processing group 1 PG(3,5), x = 5, u in L: [0, 0, 0, 0, 0, 0, ] [0, 0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [5, 5, 5, 5, 5, 5, ] PG(3,5), x = 5, u not in L: [0, 0, 0, 0, 0, 1, ] [0, 0, 0, 0, 0, 1, ] [0, 0, 0, 0, 0, 1, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [2, 2, 2, 2, 2, 3, ] #{Class patterns} = 1 #{Non-class patterns} = 1 RowSums: [1, 7, 13, 31, ] ColSums: [4, 10, ] Sle with plane and point types equations 0 0 = 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 6 0 = 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 5 = 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 = 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 20 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 5 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 30 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 = 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 = 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 = 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 | 0 0 3 = 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 = 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 = 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 6 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 4 6 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 2 3 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 2 2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 2 2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 155 | 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 651 | 0 60 -3 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 -10 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 48 -4 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 -10 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 18 -3 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 Row-reduced echelon form: 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 22 Rank: 2 The number of unknowns z_i: 2. The number of indeterminates: 20 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 = 156 4 10 = 930 6 45 = 4185 Top bound on feasible weight distribution on planes (without Zero equations): 105 51 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 = 105 0 1 = 51 0 0 = 1 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 = 0 0 1 0 0 0 = 0 0 0 1 0 0 = 0 0 0 0 1 0 = 0 0 0 0 0 1 = 0 1 0 0 0 0 = 105 0 1 0 0 0 = 51 Row-reduced echelon form of the extended system: 1 0 0 0 0 = 0 0 1 0 0 0 = 0 0 0 1 0 0 = 0 0 0 0 1 0 = 0 0 0 0 0 1 = 0 0 0 0 0 0 = 105 0 1 0 0 0 = 51 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 = 156 1 7 13 31 = 930 0 21 78 465 = 4185 Top bound on feasible weight distribution on points (without Zero equations): 68 132 41 4 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 6 = 68 0 1 0 -15 = 47 0 0 1 10 = 41 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 6 0 0 0 0 0 0 0 0 0 0 = 68 0 1 0 -15 0 0 0 0 0 0 0 0 0 0 = 47 0 0 1 10 0 0 0 0 0 0 0 0 0 0 = 41 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 = 68 0 0 0 -15 0 0 0 0 0 0 0 0 0 0 = 47 0 0 1 10 0 0 0 0 0 0 0 0 0 0 = 41 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 5 x = 6 Group 1 Possible row sums: [3, 9, 15, 21, 27, ] Possible col sums: [3, 9, 15, 21, 27, ] Processing group 1 PG(3,5), x = 6, u in L: [0, 0, 0, 0, 1, 1, ] [0, 0, 0, 0, 1, 1, ] [0, 0, 0, 0, 1, 1, ] [1, 1, 1, 1, 2, 2, ] [3, 3, 3, 3, 4, 4, ] [4, 4, 4, 4, 5, 5, ] [0, 0, 0, 1, 3, 4, ] [0, 0, 0, 1, 3, 4, ] [0, 0, 0, 1, 3, 4, ] [0, 0, 0, 1, 3, 4, ] [1, 1, 1, 2, 4, 5, ] [1, 1, 1, 2, 4, 5, ] PG(3,5), x = 6, u not in L: [0, 0, 0, 0, 1, 2, ] [0, 0, 0, 0, 1, 2, ] [0, 0, 0, 0, 1, 2, ] [1, 1, 1, 1, 2, 3, ] [1, 1, 1, 1, 2, 3, ] [1, 1, 1, 1, 2, 3, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [1, 1, 1, 2, 2, 2, ] [2, 2, 2, 3, 3, 3, ] #{Class patterns} = 2 #{Non-class patterns} = 2 RowSums: [3, 9, 15, 21, 27, ] ColSums: [3, 9, 15, 21, 27, ] Sle with plane and point types equations 0 3 0 0 = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 4 1 0 0 = 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 0 = 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 = 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 = 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 4 3 = 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 3 = 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 = 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 6 0 -9 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 8 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 4 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 Row-reduced echelon form: 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 45 Rank: 4 The number of unknowns z_i: 4. The number of indeterminates: 41 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 156 3 9 15 21 27 = 1116 3 36 105 210 351 = 5580 Top bound on feasible weight distribution on planes (without Zero equations): 91 124 43 14 7 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 91 0 1 0 -3 -8 = 22 0 0 1 3 6 = 43 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 91 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 43 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 91 0 0 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 43 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 = 156 3 9 15 21 27 = 1116 3 36 105 210 351 = 5580 Top bound on feasible weight distribution on points (without Zero equations): 91 124 43 14 7 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 91 0 1 0 -3 -8 = 22 0 0 1 3 6 = 43 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 91 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 43 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 91 0 0 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 22 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 43 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 5 x = 7 No solutions of the modular equation No feasible patterns *********************** q = 5 x = 8 Group 1 Possible row sums: [4, 10, 16, 22, 28, ] Possible col sums: [4, 10, 16, 22, 28, ] Processing group 1 PG(3,5), x = 8, u in L: [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [3, 3, 3, 4, 4, 4, ] [3, 3, 3, 4, 4, 4, ] [3, 3, 3, 4, 4, 4, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [1, 1, 1, 2, 2, 2, ] [1, 1, 1, 2, 2, 2, ] [3, 3, 3, 4, 4, 4, ] [4, 4, 4, 5, 5, 5, ] [0, 0, 0, 3, 3, 3, ] [0, 0, 0, 3, 3, 3, ] [0, 0, 0, 3, 3, 3, ] [1, 1, 1, 4, 4, 4, ] [1, 1, 1, 4, 4, 4, ] [1, 1, 1, 4, 4, 4, ] [0, 0, 1, 1, 3, 4, ] [0, 0, 1, 1, 3, 4, ] [0, 0, 1, 1, 3, 4, ] [1, 1, 2, 2, 4, 5, ] [1, 1, 2, 2, 4, 5, ] [1, 1, 2, 2, 4, 5, ] PG(3,5), x = 8, u not in L: [0, 0, 0, 0, 2, 2, ] [0, 0, 0, 0, 2, 2, ] [1, 1, 1, 1, 3, 3, ] [1, 1, 1, 1, 3, 3, ] [1, 1, 1, 1, 3, 3, ] [1, 1, 1, 1, 3, 3, ] [0, 0, 0, 1, 1, 2, ] [0, 0, 0, 1, 1, 2, ] [0, 0, 0, 1, 1, 2, ] [1, 1, 1, 2, 2, 3, ] [1, 1, 1, 2, 2, 3, ] [2, 2, 2, 3, 3, 4, ] [0, 0, 1, 1, 1, 1, ] [0, 0, 1, 1, 1, 1, ] [0, 0, 1, 1, 1, 1, ] [0, 0, 1, 1, 1, 1, ] [2, 2, 3, 3, 3, 3, ] [2, 2, 3, 3, 3, 3, ] #{Class patterns} = 4 #{Non-class patterns} = 3 RowSums: [4, 10, 16, 22, 28, ] ColSums: [4, 10, 16, 22, 28, ] Sle with plane and point types equations 0 0 3 2 0 0 0 = 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 3 0 2 0 0 0 = 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 3 0 0 0 0 0 = 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 3 1 0 0 0 = 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 0 0 0 = 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 4 3 2 = 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 2 4 = 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 2 1 0 = 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 6 2 12 6 2 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 4 0 6 8 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 8 3 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 9 2 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 2 0 0 0 = 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 45 30 -16 -6 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 24 0 24 0 -8 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 18 0 0 -24 -6 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 18 6 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 3 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 36 24 0 -6 -8 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 18 0 -6 -24 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 -4 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 9 3 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 48 Rank: 7 The number of unknowns z_i: 7. The number of indeterminates: 41 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 156 4 10 16 22 28 = 1488 6 45 120 231 378 = 8928 Top bound on feasible weight distribution on planes (without Zero equations): 78 148 66 22 11 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 78 0 1 0 -3 -8 = 12 0 0 1 3 6 = 66 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 78 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 12 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 66 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 78 0 0 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 12 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 66 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 = 156 4 10 16 22 28 = 1488 6 45 120 231 378 = 8928 Top bound on feasible weight distribution on points (without Zero equations): 78 148 66 22 11 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 78 0 1 0 -3 -8 = 12 0 0 1 3 6 = 66 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 78 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 12 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 66 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 78 0 0 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 12 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 66 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 5 x = 9 Group 1 Possible row sums: [0, 6, 12, 18, 24, 30, ] Possible col sums: [3, 9, 15, 21, 27, ] Processing group 1 PG(3,5), x = 9, u in L: [0, 0, 0, 0, 2, 3, ] [0, 0, 0, 0, 2, 3, ] [2, 2, 2, 2, 4, 5, ] [2, 2, 2, 2, 4, 5, ] [2, 2, 2, 2, 4, 5, ] [2, 2, 2, 2, 4, 5, ] [0, 0, 0, 1, 2, 2, ] [0, 0, 0, 1, 2, 2, ] [1, 1, 1, 2, 3, 3, ] [1, 1, 1, 2, 3, 3, ] [3, 3, 3, 4, 5, 5, ] [3, 3, 3, 4, 5, 5, ] [0, 0, 1, 1, 1, 2, ] [0, 0, 1, 1, 1, 2, ] [0, 0, 1, 1, 1, 2, ] [2, 2, 3, 3, 3, 4, ] [3, 3, 4, 4, 4, 5, ] [3, 3, 4, 4, 4, 5, ] [0, 0, 2, 2, 3, 4, ] [0, 0, 2, 2, 3, 4, ] [0, 0, 2, 2, 3, 4, ] [0, 0, 2, 2, 3, 4, ] [1, 1, 3, 3, 4, 5, ] [1, 1, 3, 3, 4, 5, ] [0, 0, 2, 3, 3, 3, ] [0, 0, 2, 3, 3, 3, ] [0, 0, 2, 3, 3, 3, ] [0, 0, 2, 3, 3, 3, ] [0, 0, 2, 3, 3, 3, ] [2, 2, 4, 5, 5, 5, ] [0, 1, 1, 1, 1, 1, ] [0, 1, 1, 1, 1, 1, ] [0, 1, 1, 1, 1, 1, ] [2, 3, 3, 3, 3, 3, ] [2, 3, 3, 3, 3, 3, ] [4, 5, 5, 5, 5, 5, ] PG(3,5), x = 9, u not in L: [0, 0, 0, 0, 0, 0, ] [0, 0, 0, 0, 0, 0, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [3, 3, 3, 3, 3, 3, ] [0, 0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [3, 3, 3, 3, 3, 3, ] [3, 3, 3, 3, 3, 3, ] [0, 0, 0, 2, 2, 2, ] [0, 0, 0, 2, 2, 2, ] [0, 0, 0, 2, 2, 2, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [0, 0, 1, 1, 1, 3, ] [0, 0, 1, 1, 1, 3, ] [0, 0, 1, 1, 1, 3, ] [1, 1, 2, 2, 2, 4, ] [1, 1, 2, 2, 2, 4, ] [1, 1, 2, 2, 2, 4, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [1, 1, 2, 2, 3, 3, ] [2, 2, 3, 3, 4, 4, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [4, 4, 4, 4, 4, 4, ] #{Class patterns} = 6 #{Non-class patterns} = 6 RowSums: [0, 6, 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 The number of equations: 60 Rank: 12 The number of unknowns z_i: 12. The number of indeterminates: 48 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 156 3 9 15 21 27 = 1674 3 36 105 210 351 = 10881 Top bound on feasible weight distribution on planes (without Zero equations): 60 156 103 35 17 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 60 0 1 0 -3 -8 = -9 0 0 1 3 6 = 105 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 60 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -9 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 105 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 105 The system of equations from Lemma 3.1 (feasible weight distributions of planes) no solvable The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 1 = 156 0 6 12 18 24 30 = 1674 0 15 66 153 276 435 = 10881 Top bound on feasible weight distribution on points (without Zero equations): 63 156 139 62 21 10 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 63 0 1 0 -3 -8 -15 = -93 0 0 1 3 6 10 = 186 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 63 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -93 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 186 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -93 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 186 The system of equations from Lemma 3.1 (feasible weight distributions of points) no solvable *********************** q = 5 x = 10 Group 1 Possible row sums: [1, 7, 13, 19, 25, 31, ] Possible col sums: [3, 9, 15, 21, 27, ] Processing group 1 PG(3,5), x = 10, u in L: [0, 0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [3, 3, 3, 3, 3, 3, ] [4, 4, 4, 4, 4, 4, ] [4, 4, 4, 4, 4, 4, ] [0, 0, 0, 0, 0, 0, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [3, 3, 3, 3, 3, 3, ] [5, 5, 5, 5, 5, 5, ] [0, 0, 0, 1, 2, 3, ] [0, 0, 0, 1, 2, 3, ] [2, 2, 2, 3, 4, 5, ] [2, 2, 2, 3, 4, 5, ] [2, 2, 2, 3, 4, 5, ] [2, 2, 2, 3, 4, 5, ] [0, 0, 0, 2, 2, 2, ] [0, 0, 0, 2, 2, 2, ] [1, 1, 1, 3, 3, 3, ] [2, 2, 2, 4, 4, 4, ] [2, 2, 2, 4, 4, 4, ] [3, 3, 3, 5, 5, 5, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [1, 1, 2, 2, 3, 3, ] [1, 1, 2, 2, 3, 3, ] [3, 3, 4, 4, 5, 5, ] [3, 3, 4, 4, 5, 5, ] [0, 0, 3, 3, 3, 3, ] [0, 0, 3, 3, 3, 3, ] [0, 0, 3, 3, 3, 3, ] [0, 0, 3, 3, 3, 3, ] [1, 1, 4, 4, 4, 4, ] [1, 1, 4, 4, 4, 4, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [2, 3, 3, 3, 3, 4, ] [3, 4, 4, 4, 4, 5, ] [3, 4, 4, 4, 4, 5, ] [0, 1, 1, 3, 3, 4, ] [0, 1, 1, 3, 3, 4, ] [0, 1, 1, 3, 3, 4, ] [0, 1, 1, 3, 3, 4, ] [1, 2, 2, 4, 4, 5, ] [1, 2, 2, 4, 4, 5, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [3, 3, 3, 3, 3, 3, ] [3, 3, 3, 3, 3, 3, ] [5, 5, 5, 5, 5, 5, ] [1, 1, 1, 1, 4, 4, ] [1, 1, 1, 1, 4, 4, ] [1, 1, 1, 1, 4, 4, ] [1, 1, 1, 1, 4, 4, ] [2, 2, 2, 2, 5, 5, ] [2, 2, 2, 2, 5, 5, ] PG(3,5), x = 10, u not in L: [0, 0, 0, 0, 0, 1, ] [0, 0, 0, 0, 0, 1, ] [2, 2, 2, 2, 2, 3, ] [2, 2, 2, 2, 2, 3, ] [2, 2, 2, 2, 2, 3, ] [3, 3, 3, 3, 3, 4, ] [0, 0, 0, 0, 0, 1, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [3, 3, 3, 3, 3, 4, ] [3, 3, 3, 3, 3, 4, ] [0, 0, 1, 1, 2, 3, ] [0, 0, 1, 1, 2, 3, ] [0, 0, 1, 1, 2, 3, ] [1, 1, 2, 2, 3, 4, ] [1, 1, 2, 2, 3, 4, ] [1, 1, 2, 2, 3, 4, ] [0, 0, 1, 2, 2, 2, ] [0, 0, 1, 2, 2, 2, ] [0, 0, 1, 2, 2, 2, ] [0, 0, 1, 2, 2, 2, ] [1, 1, 2, 3, 3, 3, ] [2, 2, 3, 4, 4, 4, ] [0, 1, 1, 1, 1, 3, ] [0, 1, 1, 1, 1, 3, ] [0, 1, 1, 1, 1, 3, ] [0, 1, 1, 1, 1, 3, ] [1, 2, 2, 2, 2, 4, ] [2, 3, 3, 3, 3, 5, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [1, 1, 1, 1, 1, 2, ] [4, 4, 4, 4, 4, 5, ] #{Class patterns} = 10 #{Non-class patterns} = 6 RowSums: [1, 7, 13, 19, 25, 31, ] ColSums: [3, 9, 15, 21, 27, ] Sle with plane and point types equations 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 3 3 2 0 1 2 0 4 0 0 0 0 0 0 = 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 6 6 1 0 2 0 4 0 6 0 0 0 0 0 0 0 = 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 3 2 4 1 2 0 0 0 0 0 0 0 0 = 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 1 0 0 0 0 1 0 2 0 0 0 0 0 0 = 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 = 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 5 5 2 1 4 5 = 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -3200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1280 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1560 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -640 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -380 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -320 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -240 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 The number of equations: 62 Rank: 16 The number of unknowns z_i: 16. The number of indeterminates: 48 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 156 3 9 15 21 27 = 1860 3 36 105 210 351 = 13020 Top bound on feasible weight distribution on planes (without Zero equations): 60 156 124 45 20 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 60 0 1 0 -3 -8 = -40 0 0 1 3 6 = 136 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 20 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 80 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 16 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 = 380 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 = 1600 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 320 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -2 = 800 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -4 = 6320 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 1280 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 = 3200 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 240 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 640 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -4 = 1560 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 = 0 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 60 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -40 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 136 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 20 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 80 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 16 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 = 380 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 = 1600 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 320 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -2 = 800 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -4 = 6320 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 1280 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 = 3200 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -2 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 240 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 640 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -4 = 1560 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 = 0 Top bound on feasible weight distribution on planes: 20 80 16 40 0 Feasible weight distribution of planes 1: [20, 80, 16, 40, 0, ] The number of solutions of the equations from Lemma 3.1: 2 The number of solutions of the extended system:1 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 1 = 156 1 7 13 19 25 31 = 1860 0 21 78 171 300 465 = 13020 Top bound on feasible weight distribution on points (without Zero equations): 68 156 143 65 22 11 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 68 0 1 0 -3 -8 -15 = -108 0 0 1 3 6 10 = 196 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 80 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 40 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 20 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 6320 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 3200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 1600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 1280 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 1560 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = 800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 = 640 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 = 380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 = 320 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 = 240 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 68 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -108 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 196 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 80 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 40 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 20 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 6320 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 3200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 = 1600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 = 1280 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 = 1560 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 0 0 0 160 120 30 0 0 0 0 0 0 160 0 240 96 Solutions count: 1 Solutions: [0, 0, 0, 160, 120, 30, 0, 0, 0, 0, 0, 0, 160, 0, 240, 96, ] Total solutions count: 1 *********************** q = 5 x = 11 No solutions of the modular equation No feasible patterns *********************** q = 5 x = 12 Group 1 Possible row sums: [0, 6, 12, 18, 24, 30, ] Possible col sums: [0, 6, 12, 18, 24, 30, ] Processing group 1 PG(3,5), x = 12, u in L: [0, 0, 0, 1, 2, 2, ] [1, 1, 1, 2, 3, 3, ] [2, 2, 2, 3, 4, 4, ] [2, 2, 2, 3, 4, 4, ] [3, 3, 3, 4, 5, 5, ] [3, 3, 3, 4, 5, 5, ] [0, 0, 1, 1, 1, 2, ] [1, 1, 2, 2, 2, 3, ] [1, 1, 2, 2, 2, 3, ] [3, 3, 4, 4, 4, 5, ] [3, 3, 4, 4, 4, 5, ] [3, 3, 4, 4, 4, 5, ] [0, 0, 2, 3, 3, 3, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [0, 1, 1, 1, 1, 1, ] [0, 1, 1, 1, 1, 1, ] [2, 3, 3, 3, 3, 3, ] [3, 4, 4, 4, 4, 4, ] [3, 4, 4, 4, 4, 4, ] [3, 4, 4, 4, 4, 4, ] [0, 1, 1, 1, 1, 1, ] [1, 2, 2, 2, 2, 2, ] [1, 2, 2, 2, 2, 2, ] [2, 3, 3, 3, 3, 3, ] [3, 4, 4, 4, 4, 4, ] [4, 5, 5, 5, 5, 5, ] [0, 1, 1, 2, 3, 4, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [0, 1, 1, 3, 3, 3, ] [0, 1, 1, 3, 3, 3, ] [1, 2, 2, 4, 4, 4, ] [1, 2, 2, 4, 4, 4, ] [1, 2, 2, 4, 4, 4, ] [2, 3, 3, 5, 5, 5, ] [0, 1, 2, 2, 3, 3, ] [0, 1, 2, 2, 3, 3, ] [0, 1, 2, 2, 3, 3, ] [1, 2, 3, 3, 4, 4, ] [2, 3, 4, 4, 5, 5, ] [2, 3, 4, 4, 5, 5, ] PG(3,5), x = 12, u not in L: [0, 0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [3, 3, 3, 3, 3, 3, ] [4, 4, 4, 4, 4, 4, ] [0, 0, 0, 1, 2, 3, ] [1, 1, 1, 2, 3, 4, ] [1, 1, 1, 2, 3, 4, ] [1, 1, 1, 2, 3, 4, ] [1, 1, 1, 2, 3, 4, ] [2, 2, 2, 3, 4, 5, ] [0, 0, 0, 2, 2, 2, ] [0, 0, 0, 2, 2, 2, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [2, 2, 2, 4, 4, 4, ] [2, 2, 2, 4, 4, 4, ] [0, 0, 1, 1, 1, 3, ] [0, 0, 1, 1, 1, 3, ] [1, 1, 2, 2, 2, 4, ] [1, 1, 2, 2, 2, 4, ] [2, 2, 3, 3, 3, 5, ] [2, 2, 3, 3, 3, 5, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [2, 2, 3, 3, 4, 4, ] [2, 2, 3, 3, 4, 4, ] [2, 2, 3, 3, 4, 4, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [1, 1, 2, 2, 3, 3, ] [1, 1, 2, 2, 3, 3, ] [1, 1, 2, 2, 3, 3, ] [3, 3, 4, 4, 5, 5, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [1, 2, 2, 2, 2, 3, ] [2, 3, 3, 3, 3, 4, ] [3, 4, 4, 4, 4, 5, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] #{Class patterns} = 8 #{Non-class patterns} = 8 RowSums: [0, 6, 12, 18, 24, 30, ] ColSums: [0, 6, 12, 18, 24, 30, ] Sle with plane and point types equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 -817 0 0 137082 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 -56730 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 -324012 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 The number of equations: 69 Rank: 16 The number of unknowns z_i: 16. The number of indeterminates: 55 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 1 = 156 0 6 12 18 24 30 = 2232 0 15 66 153 276 435 = 17856 Top bound on feasible weight distribution on planes (without Zero equations): 125 156 156 113 41 20 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 125 0 1 0 -3 -8 -15 = -310 0 0 1 3 6 10 = 341 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 1 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 14694 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 2046 0 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 7998 0 0 0 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 5394 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 62 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 0 0 0 79 0 0 = 148800 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 0 0 0 0 0 0 0 0 0 -79 0 0 = 237894 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 0 0 0 0 0 613 0 0 = 881454 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 1 0 0 = 78771 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 -107 0 0 = -31434 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 16 0 0 = 32178 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 -817 0 0 = -137082 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 = 56730 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 = 324012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 125 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -310 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 341 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 1 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 14694 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 2046 0 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 7998 0 0 0 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 5394 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 62 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 0 0 0 79 0 0 = 148800 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 0 0 0 0 0 0 0 0 0 -79 0 0 = 237894 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 0 0 0 0 0 613 0 0 = 881454 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 1 0 0 = 78771 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 -107 0 0 = -31434 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 16 0 0 = 32178 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 -817 0 0 = -137082 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 = 56730 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 = 324012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on planes: 1 31 62 31 31 0 Feasible weight distribution of planes 1: [1, 31, 62, 31, 31, 0, ] The number of solutions of the equations from Lemma 3.1: 2 The number of solutions of the extended system:1 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 1 = 156 0 6 12 18 24 30 = 2232 0 15 66 153 276 435 = 17856 Top bound on feasible weight distribution on points (without Zero equations): 125 156 156 113 41 20 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 125 0 1 0 -3 -8 -15 = -310 0 0 1 3 6 10 = 341 The extended system (The equations from Lemma 3.1 and the Zero equations): 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 1 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 14694 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 2046 0 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 7998 0 0 0 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 5394 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 62 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 0 0 0 79 0 0 = 148800 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 0 0 0 0 0 0 0 0 0 -79 0 0 = 237894 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 0 0 0 0 0 613 0 0 = 881454 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 1 0 0 = 78771 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 -107 0 0 = -31434 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 16 0 0 = 32178 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 -817 0 0 = -137082 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 = 56730 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 = 324012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 125 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -310 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 341 Row-reduced echelon form of the extended system: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 1 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 14694 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 2046 0 0 0 108 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 = 7998 0 0 0 0 324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 0 = 5394 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 62 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 31 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 0 0 0 79 0 0 = 148800 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 0 0 0 0 0 0 0 0 0 -79 0 0 = 237894 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 0 0 0 0 0 613 0 0 = 881454 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 81 0 0 0 0 0 0 0 0 1 0 0 = 78771 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 -107 0 0 = -31434 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 16 0 0 = 32178 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 324 0 0 0 0 -817 0 0 = -137082 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 = 56730 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 = 324012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on points: 1 31 62 31 31 0 Feasible weight distribution of points 1: [1, 31, 62, 31, 31, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 1: [1, 31, 62, 31, 31, 0, ] Feasible weight distribution of points 1: [1, 31, 62, 31, 31, 0, ] 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8 0 0 9951 | 2511 0 54 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 -1488 | 5022 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 930 | 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -930 | 0 0 0 0 0 1 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 43 0 0 -56730 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 162 0 181 0 0 -324012 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 93 93 0 0 0 0 93 93 31 93 0 93 0 93 93 31 Solutions count: 1 Solutions: [93, 93, 0, 0, 0, 0, 93, 93, 31, 93, 0, 93, 0, 93, 93, 31, ] Total solutions count: 1 *********************** q = 5 x = 13 Group 1 Possible row sums: [0, 6, 12, 18, 24, 30, ] Possible col sums: [1, 7, 13, 19, 25, 31, ] Group 2 Possible row sums: [3, 9, 15, 21, 27, ] Possible col sums: [4, 10, 16, 22, 28, ] Processing group 1 PG(3,5), x = 13, u in L: [0, 0, 0, 1, 2, 2, ] [2, 2, 2, 3, 4, 4, ] [2, 2, 2, 3, 4, 4, ] [2, 2, 2, 3, 4, 4, ] [3, 3, 3, 4, 5, 5, ] [3, 3, 3, 4, 5, 5, ] [0, 0, 1, 1, 1, 2, ] [1, 1, 2, 2, 2, 3, ] [2, 2, 3, 3, 3, 4, ] [3, 3, 4, 4, 4, 5, ] [3, 3, 4, 4, 4, 5, ] [3, 3, 4, 4, 4, 5, ] [0, 1, 1, 1, 1, 1, ] [1, 2, 2, 2, 2, 2, ] [2, 3, 3, 3, 3, 3, ] [2, 3, 3, 3, 3, 3, ] [3, 4, 4, 4, 4, 4, ] [4, 5, 5, 5, 5, 5, ] [0, 1, 1, 3, 3, 3, ] [1, 2, 2, 4, 4, 4, ] [1, 2, 2, 4, 4, 4, ] [1, 2, 2, 4, 4, 4, ] [1, 2, 2, 4, 4, 4, ] [2, 3, 3, 5, 5, 5, ] [0, 1, 2, 2, 3, 3, ] [0, 1, 2, 2, 3, 3, ] [1, 2, 3, 3, 4, 4, ] [1, 2, 3, 3, 4, 4, ] [2, 3, 4, 4, 5, 5, ] [2, 3, 4, 4, 5, 5, ] [0, 2, 2, 2, 2, 3, ] [0, 2, 2, 2, 2, 3, ] [0, 2, 2, 2, 2, 3, ] [2, 4, 4, 4, 4, 5, ] [2, 4, 4, 4, 4, 5, ] [2, 4, 4, 4, 4, 5, ] [0, 3, 3, 3, 4, 4, ] [0, 3, 3, 3, 4, 4, ] [0, 3, 3, 3, 4, 4, ] [0, 3, 3, 3, 4, 4, ] [0, 3, 3, 3, 4, 4, ] [0, 3, 3, 3, 4, 4, ] [1, 1, 1, 2, 3, 3, ] [1, 1, 1, 2, 3, 3, ] [1, 1, 1, 2, 3, 3, ] [3, 3, 3, 4, 5, 5, ] [3, 3, 3, 4, 5, 5, ] [3, 3, 3, 4, 5, 5, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 1, 3, 4, 4, 4, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] [1, 2, 2, 3, 4, 5, ] PG(3,5), x = 13, u not in L: [0, 0, 0, 0, 0, 0, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [3, 3, 3, 3, 3, 3, ] [3, 3, 3, 3, 3, 3, ] [4, 4, 4, 4, 4, 4, ] [0, 0, 0, 1, 2, 3, ] [1, 1, 1, 2, 3, 4, ] [1, 1, 1, 2, 3, 4, ] [1, 1, 1, 2, 3, 4, ] [2, 2, 2, 3, 4, 5, ] [2, 2, 2, 3, 4, 5, ] [0, 0, 0, 2, 2, 2, ] [0, 0, 0, 2, 2, 2, ] [1, 1, 1, 3, 3, 3, ] [2, 2, 2, 4, 4, 4, ] [2, 2, 2, 4, 4, 4, ] [2, 2, 2, 4, 4, 4, ] [0, 0, 0, 2, 2, 2, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [1, 1, 1, 3, 3, 3, ] [3, 3, 3, 5, 5, 5, ] [0, 0, 1, 1, 1, 3, ] [0, 0, 1, 1, 1, 3, ] [1, 1, 2, 2, 2, 4, ] [2, 2, 3, 3, 3, 5, ] [2, 2, 3, 3, 3, 5, ] [2, 2, 3, 3, 3, 5, ] [0, 0, 1, 1, 2, 2, ] [0, 0, 1, 1, 2, 2, ] [1, 1, 2, 2, 3, 3, ] [1, 1, 2, 2, 3, 3, ] [2, 2, 3, 3, 4, 4, ] [3, 3, 4, 4, 5, 5, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [0, 1, 1, 1, 1, 2, ] [2, 3, 3, 3, 3, 4, ] [2, 3, 3, 3, 3, 4, ] [3, 4, 4, 4, 4, 5, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [0, 1, 2, 2, 3, 4, ] [1, 2, 3, 3, 4, 5, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [4, 4, 4, 4, 4, 4, ] [4, 4, 4, 4, 4, 4, ] [1, 1, 1, 1, 1, 1, ] [1, 1, 1, 1, 1, 1, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [2, 2, 2, 2, 2, 2, ] [5, 5, 5, 5, 5, 5, ] #{Class patterns} = 10 #{Non-class patterns} = 10 RowSums: [0, 6, 12, 18, 24, 30, ] ColSums: [1, 7, 13, 19, 25, 31, ] Sle with plane and point types equations 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 1 1 1 0 0 2 1 0 0 0 0 0 0 0 0 0 0 = 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 3 2 1 2 1 0 0 3 0 2 0 0 0 0 0 0 0 0 0 0 = 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 3 5 0 2 4 3 1 1 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 2 1 0 3 2 1 2 2 3 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 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0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7936 0 0 0 0 0 0 0 266752 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18635 160468 0 242265 1627083 0 -642486780 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7936 0 0 0 0 0 0 -37376 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3685 -11668 0 3255 -65179 0 -8632260 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7936 0 0 0 0 0 150016 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7535 63556 0 99045 672879 0 -262667340 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1240 0 0 0 -30000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2605 -12620 0 -25575 -159693 0 57669300 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 238080 0 0 -990720 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -51965 -713740 0 -959295 -7017981 0 2262401700 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119040 1175040 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 33685 303020 0 486855 3445909 0 -1309709700 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15872 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119040 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1225 -6620 0 -8835 -58441 0 21882900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 29760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345 1500 0 1395 9689 0 -6407700 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -205 -460 0 465 51 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -4 0 -21 1 0 -24180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 256 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3720 0 0 0 0 0 0 0 0 -2605 -12620 0 -18135 -116293 0 47513700 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1860 0 0 0 0 0 0 0 365 -920 0 -465 -7617 0 -967200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 0 0 0 0 -463 -1668 0 -2325 -15135 0 6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 -395 -3028 0 -3993 -23435 0 9551100 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 -16 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 12265 173660 0 53475 745449 0 -466794900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 463 676 0 2325 10175 0 -6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 -9599460 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 The number of equations: 69 Rank: 20 The number of unknowns z_i: 20. The number of indeterminates: 55 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 1 = 156 1 7 13 19 25 31 = 2418 0 21 78 171 300 465 = 20553 Top bound on feasible weight distribution on planes (without Zero equations): 130 156 156 117 43 21 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 130 0 1 0 -3 -8 -15 = -325 0 0 1 3 6 10 = 351 The extended system (The equations from Lemma 3.1 and the Zero equations): 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3937 23808 0 35991 193902 0 = 66023490 0 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12803 -69378 0 -92907 -483420 0 = -148295940 0 0 113522 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1643 7254 0 6975 35534 0 = 23125752 0 0 0 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6913 -17670 0 9207 2082 0 = 16889730 0 0 0 0 510849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 -1023 0 -7533 294 0 = 8317920 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 170283 0 0 0 0 0 0 0 0 0 0 0 0 0 19685 119040 0 179955 969510 0 = 330117450 0 0 0 0 0 0 0 0 170283 0 0 0 0 0 0 0 0 0 0 0 0 39370 238080 0 359910 1939020 0 = 660234900 0 0 0 0 0 0 0 0 0 340566 0 0 0 0 0 0 0 0 0 0 0 43307 261888 0 395901 2132922 0 = 726258390 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 0 0 0 0 62992 380928 0 575856 3102432 0 = 1056375840 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 0 0 -429629 -2083200 0 -2893788 -14906550 0 = -5079746490 0 0 0 0 0 0 0 0 0 0 0 0 0 340566 0 0 0 0 0 0 0 -29915 -800358 0 -1110141 -6134874 0 = -1365009360 0 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 -419554 -1241922 0 -1482327 -7997937 0 = -2719246530 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 -157232 -1208256 0 -1698831 -7755933 0 = -2636780640 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 56761 0 0 0 63550 486204 0 375813 2056804 0 = 1154280660 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 170283 0 0 109895 -3162 0 534006 1823586 0 = 979604340 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 170283 0 57040 393204 0 155403 1736640 0 = 956560800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 130 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -325 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 351 Row-reduced echelon form of the extended system: 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3937 23808 0 35991 193902 0 = 66023490 0 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12803 -69378 0 -92907 -483420 0 = -148295940 0 0 113522 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1643 7254 0 6975 35534 0 = 23125752 0 0 0 1021698 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6913 -17670 0 9207 2082 0 = 16889730 0 0 0 0 510849 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 -1023 0 -7533 294 0 = 8317920 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 170283 0 0 0 0 0 0 0 0 0 0 0 0 0 19685 119040 0 179955 969510 0 = 330117450 0 0 0 0 0 0 0 0 170283 0 0 0 0 0 0 0 0 0 0 0 0 39370 238080 0 359910 1939020 0 = 660234900 0 0 0 0 0 0 0 0 0 340566 0 0 0 0 0 0 0 0 0 0 0 43307 261888 0 395901 2132922 0 = 726258390 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 0 0 0 0 62992 380928 0 575856 3102432 0 = 1056375840 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 0 0 -429629 -2083200 0 -2893788 -14906550 0 = -5079746490 0 0 0 0 0 0 0 0 0 0 0 0 0 340566 0 0 0 0 0 0 0 -29915 -800358 0 -1110141 -6134874 0 = -1365009360 0 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 0 -419554 -1241922 0 -1482327 -7997937 0 = -2719246530 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 510849 0 0 0 0 0 -157232 -1208256 0 -1698831 -7755933 0 = -2636780640 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 56761 0 0 0 63550 486204 0 375813 2056804 0 = 1154280660 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 170283 0 0 109895 -3162 0 534006 1823586 0 = 979604340 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 170283 0 57040 393204 0 155403 1736640 0 = 956560800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on planes: 1 52 106 117 43 1 Feasible weight distribution of planes 1: [0, 32, 60, 31, 33, 0, ] Feasible weight distribution of planes 2: [0, 28, 72, 19, 37, 0, ] Feasible weight distribution of planes 3: [0, 31, 62, 31, 31, 1, ] The number of solutions of the equations from Lemma 3.1: 127 The number of solutions of the extended system:3 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 1 = 156 0 6 12 18 24 30 = 2418 0 15 66 153 276 435 = 20553 Top bound on feasible weight distribution on points (without Zero equations): 156 156 156 134 52 26 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 6 = 156 0 1 0 -3 -8 -15 = -403 0 0 1 3 6 10 = 403 The extended system (The equations from Lemma 3.1 and the Zero equations): 15872 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 119040 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1225 -6620 0 -8835 -58441 0 = -21882900 0 0 29760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345 1500 0 1395 9689 0 = 6407700 0 0 0 39680 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -205 -460 0 465 51 0 = 1233180 0 0 0 0 1536 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -4 0 -21 1 0 = 24180 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 256 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 3720 0 0 0 0 0 0 0 0 -2605 -12620 0 -18135 -116293 0 = -47513700 0 0 0 0 0 0 0 0 0 0 0 0 0 1860 0 0 0 0 0 0 0 365 -920 0 -465 -7617 0 = 967200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 0 0 0 0 -463 -1668 0 -2325 -15135 0 = -6165900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 -395 -3028 0 -3993 -23435 0 = -9551100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 -16 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 12265 173660 0 53475 745449 0 = 466794900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 463 676 0 2325 10175 0 = 6165900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 = 9599460 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 1 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 156 0 1 0 -3 -8 -15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -403 0 0 1 3 6 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 403 Row-reduced echelon form of the extended system: 15872 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 119040 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1225 -6620 0 -8835 -58441 0 = -21882900 0 0 29760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345 1500 0 1395 9689 0 = 6407700 0 0 0 39680 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -205 -460 0 465 51 0 = 1233180 0 0 0 0 1536 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -4 0 -21 1 0 = 24180 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 256 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 = 1233180 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 3720 0 0 0 0 0 0 0 0 -2605 -12620 0 -18135 -116293 0 = -47513700 0 0 0 0 0 0 0 0 0 0 0 0 0 1860 0 0 0 0 0 0 0 365 -920 0 -465 -7617 0 = 967200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 0 0 0 0 -463 -1668 0 -2325 -15135 0 = -6165900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 -395 -3028 0 -3993 -23435 0 = -9551100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 -16 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 12265 173660 0 53475 745449 0 = 466794900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 463 676 0 2325 10175 0 = 6165900 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 = 9599460 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 = 0 Top bound on feasible weight distribution on points: 1 43 99 116 52 1 Feasible weight distribution of points 1: [0, 38, 16, 75, 27, 0, ] Feasible weight distribution of points 2: [0, 37, 19, 72, 28, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 2: [0, 28, 72, 19, 37, 0, ] Feasible weight distribution of points 2: [0, 37, 19, 72, 28, 0, ] 47616 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -69120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663 -11868 0 -17763 -118121 0 43393428 | 9427968 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11385 138780 0 194835 1053817 0 -479561940 | 5713920 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -92160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5895 -91620 0 -102765 -1181191 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1860 0 0 0 0 0 0 0 365 -920 0 -465 -7617 0 -967200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 0 0 0 0 -463 -1668 0 -2325 -15135 0 6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 -395 -3028 0 -3993 -23435 0 9551100 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 -16 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 12265 173660 0 53475 745449 0 -466794900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 463 676 0 2325 10175 0 -6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 -9599460 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 198 24 0 111 18 1 0 18 33 0 0 24 18 1 198 18 111 0 33 0 Solutions count: 1 Solutions: [198, 24, 0, 111, 18, 1, 0, 18, 33, 0, 0, 24, 18, 1, 198, 18, 111, 0, 33, 0, ] Feasible weight distribution of points 3: [0, 36, 22, 69, 29, 0, ] Feasible weight distribution of points 4: [0, 35, 25, 66, 30, 0, ] Feasible weight distribution of points 5: [1, 31, 31, 62, 31, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 3: [0, 31, 62, 31, 31, 1, ] Feasible weight distribution of points 5: [1, 31, 31, 62, 31, 0, ] 47616 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -69120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663 -11868 0 -17763 -118121 0 43393428 | 4428288 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11385 138780 0 194835 1053817 0 -479561940 | 22141440 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -92160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5895 -91620 0 -102765 -1181191 0 318958380 | 22141440 0 0 0 0 79360 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 -9599460 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 93 93 0 93 93 0 0 0 0 31 31 93 0 0 93 93 93 0 0 0 Solutions count: 1 Solutions: [93, 93, 0, 93, 93, 0, 0, 0, 0, 31, 31, 93, 0, 0, 93, 93, 93, 0, 0, 0, ] Feasible weight distribution of points 6: [0, 34, 28, 63, 31, 0, ] Feasible weight distribution of points 7: [0, 33, 31, 60, 32, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 1: [0, 32, 60, 31, 33, 0, ] Feasible weight distribution of points 7: [0, 33, 31, 60, 32, 0, ] 47616 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -69120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663 -11868 0 -17763 -118121 0 43393428 | 5332992 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 345600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11385 138780 0 194835 1053817 0 -479561940 | 20474880 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | 0 0 0 0 238080 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -92160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5895 -91620 0 -102765 -1181191 0 318958380 | 20236800 0 0 0 0 79360 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 69120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1755 45420 0 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0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 256 0 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 512 0 0 0 0 0 0 0 0 0 0 51 308 0 465 3123 0 -1233180 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3720 0 0 0 0 0 0 0 0 -2605 -12620 0 -18135 -116293 0 47513700 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1860 0 0 0 0 0 0 0 365 -920 0 -465 -7617 0 -967200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 0 0 0 0 -463 -1668 0 -2325 -15135 0 6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 0 0 0 0 -395 -3028 0 -3993 -23435 0 9551100 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 -16 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 39680 0 0 0 12265 173660 0 53475 745449 0 -466794900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 496 0 0 463 676 0 2325 10175 0 -6165900 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1536 0 397 3020 0 879 15757 0 -9599460 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 -11 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 | 0 Top bound on solutions: 112 86 0 85 96 1 0 2 21 0 0 86 2 1 112 96 85 0 21 0 Solutions count: 1 Solutions: [112, 86, 0, 85, 96, 1, 0, 2, 21, 0, 0, 86, 2, 1, 112, 96, 85, 0, 21, 0, ] Total solutions count: 3 Processing group 2 PG(3,5), x = 13, u in L: [0, 0, 0, 0, 0, 2, ] [3, 3, 3, 3, 3, 5, ] [3, 3, 3, 3, 3, 5, ] [3, 3, 3, 3, 3, 5, ] [3, 3, 3, 3, 3, 5, ] [3, 3, 3, 3, 3, 5, ] [0, 0, 0, 0, 1, 1, ] [2, 2, 2, 2, 3, 3, ] [3, 3, 3, 3, 4, 4, ] [3, 3, 3, 3, 4, 4, ] [3, 3, 3, 3, 4, 4, ] [4, 4, 4, 4, 5, 5, ] [0, 0, 1, 2, 2, 3, ] [1, 1, 2, 3, 3, 4, ] [2, 2, 3, 4, 4, 5, ] [2, 2, 3, 4, 4, 5, ] [2, 2, 3, 4, 4, 5, ] [2, 2, 3, 4, 4, 5, ] [0, 0, 2, 2, 2, 2, ] [1, 1, 3, 3, 3, 3, ] [1, 1, 3, 3, 3, 3, ] [2, 2, 4, 4, 4, 4, ] [2, 2, 4, 4, 4, 4, ] [3, 3, 5, 5, 5, 5, ] [0, 1, 1, 2, 2, 2, ] [0, 1, 1, 2, 2, 2, ] [2, 3, 3, 4, 4, 4, ] [2, 3, 3, 4, 4, 4, ] [2, 3, 3, 4, 4, 4, ] [3, 4, 4, 5, 5, 5, ] [0, 1, 1, 2, 2, 2, ] [1, 2, 2, 3, 3, 3, ] [1, 2, 2, 3, 3, 3, ] [1, 2, 2, 3, 3, 3, ] [3, 4, 4, 5, 5, 5, ] [3, 4, 4, 5, 5, 5, ] [0, 2, 2, 3, 3, 4, ] [0, 2, 2, 3, 3, 4, ] [0, 2, 2, 3, 3, 4, ] [1, 3, 3, 4, 4, 5, ] [1, 3, 3, 4, 4, 5, ] [1, 3, 3, 4, 4, 5, ] [0, 2, 3, 3, 3, 3, ] [0, 2, 3, 3, 3, 3, ] [0, 2, 3, 3, 3, 3, ] [0, 2, 3, 3, 3, 3, ] [1, 3, 4, 4, 4, 4, ] [2, 4, 5, 5, 5, 5, ] [1, 1, 1, 1, 2, 2, ] [1, 1, 1, 1, 2, 2, ] [2, 2, 2, 2, 3, 3, ] [3, 3, 3, 3, 4, 4, ] [4, 4, 4, 4, 5, 5, ] [4, 4, 4, 4, 5, 5, ] [1, 1, 2, 2, 4, 4, ] [1, 1, 2, 2, 4, 4, ] [1, 1, 2, 2, 4, 4, ] [2, 2, 3, 3, 5, 5, ] [2, 2, 3, 3, 5, 5, ] [2, 2, 3, 3, 5, 5, ] PG(3,5), x = 13, u not in L: [0, 0, 0, 0, 0, 3, ] [2, 2, 2, 2, 2, 5, ] [2, 2, 2, 2, 2, 5, ] [2, 2, 2, 2, 2, 5, ] [2, 2, 2, 2, 2, 5, ] [2, 2, 2, 2, 2, 5, ] [0, 0, 0, 0, 1, 2, ] [1, 1, 1, 1, 2, 3, ] [1, 1, 1, 1, 2, 3, ] [2, 2, 2, 2, 3, 4, ] [3, 3, 3, 3, 4, 5, ] [3, 3, 3, 3, 4, 5, ] [0, 0, 0, 1, 1, 1, ] [0, 0, 0, 1, 1, 1, ] [2, 2, 2, 3, 3, 3, ] [2, 2, 2, 3, 3, 3, ] [3, 3, 3, 4, 4, 4, ] [3, 3, 3, 4, 4, 4, ] [0, 0, 0, 1, 1, 1, ] [1, 1, 1, 2, 2, 2, ] [1, 1, 1, 2, 2, 2, ] [2, 2, 2, 3, 3, 3, ] [2, 2, 2, 3, 3, 3, ] [4, 4, 4, 5, 5, 5, ] [0, 0, 1, 2, 3, 3, ] [0, 0, 1, 2, 3, 3, ] [1, 1, 2, 3, 4, 4, ] [1, 1, 2, 3, 4, 4, ] [1, 1, 2, 3, 4, 4, ] [1, 1, 2, 3, 4, 4, ] [0, 0, 2, 2, 2, 3, ] [0, 0, 2, 2, 2, 3, ] [0, 0, 2, 2, 2, 3, ] [1, 1, 3, 3, 3, 4, ] [1, 1, 3, 3, 3, 4, ] [2, 2, 4, 4, 4, 5, ] [0, 1, 1, 1, 2, 4, ] [0, 1, 1, 1, 2, 4, ] [1, 2, 2, 2, 3, 5, ] [1, 2, 2, 2, 3, 5, ] [1, 2, 2, 2, 3, 5, ] [1, 2, 2, 2, 3, 5, ] [0, 1, 1, 1, 3, 3, ] [0, 1, 1, 1, 3, 3, ] [0, 1, 1, 1, 3, 3, ] [1, 2, 2, 2, 4, 4, ] [1, 2, 2, 2, 4, 4, ] [2, 3, 3, 3, 5, 5, ] [0, 1, 1, 2, 2, 3, ] [0, 1, 1, 2, 2, 3, ] [0, 1, 1, 2, 2, 3, ] [0, 1, 1, 2, 2, 3, ] [2, 3, 3, 4, 4, 5, ] [2, 3, 3, 4, 4, 5, ] [0, 1, 2, 2, 2, 2, ] [0, 1, 2, 2, 2, 2, ] [0, 1, 2, 2, 2, 2, ] [0, 1, 2, 2, 2, 2, ] [1, 2, 3, 3, 3, 3, ] [3, 4, 5, 5, 5, 5, ] #{Class patterns} = 10 #{Non-class patterns} = 10 RowSums: [3, 9, 15, 21, 27, ] ColSums: [4, 10, 16, 22, 28, ] Sle with plane and point types equations 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 = 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 0 2 2 1 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 = 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 5 4 1 0 2 2 2 1 4 2 0 0 0 0 0 0 0 0 0 0 = 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 0 2 2 4 3 3 2 4 2 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 = 0 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0 0 0 0 0 0 0 0 711 2718 2982 -4923880 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 -227 -102 1090 -1021800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 -5 -402 -1042 20800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 6641 25770 43998 -37457160 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 -2657 -6424 -13792 11053120 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 18053 63078 120386 -100660040 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 4909 40782 49050 -45981000 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 -55 -641 -119 228800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 1829 -25362 -2986 -7608640 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 -25795 -77550 -203270 107307200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 -5895 -27800 -26160 24523200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 935 -15570 -47130 -3889600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 47277 153162 215514 -259588160 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 32405 155274 280130 -262602600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 -635 -5682 -41590 2641600 | 0 The number of equations: 57 Rank: 20 The number of unknowns z_i: 20. The number of indeterminates: 41 The system of equations from Lemma 3.1 (feasible weight distributions of planes): 1 1 1 1 1 = 156 4 10 16 22 28 = 2418 6 45 120 231 378 = 20553 Top bound on feasible weight distribution on planes (without Zero equations): 78 156 151 73 26 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 78 0 1 0 -3 -8 = -143 0 0 1 3 6 = 221 The extended system (The equations from Lemma 3.1 and the Zero equations): 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 143 783 1180 = 767000 0 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -413 -2103 -2880 = -890500 0 0 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 381 1611 1560 = 1995500 0 0 0 3020 0 0 0 0 0 0 0 0 0 0 0 0 0 -19 -9 160 = 104000 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 0 0 -8 -123 -330 = -18200 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 0 3413 14253 20930 = 13604500 0 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 -3347 -6632 -8770 = -5700500 0 0 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 7724 33819 50590 = 32883500 0 0 0 0 0 0 0 0 3020 0 0 0 0 0 0 0 0 1557 8367 12320 = 8008000 0 0 0 0 0 0 0 0 0 302 0 0 0 0 0 0 0 -24 -67 -84 = -54600 0 0 0 0 0 0 0 0 0 0 3775 0 0 0 0 0 0 -4812 -34347 -43720 = -12714000 0 0 0 0 0 0 0 0 0 0 0 15100 0 0 0 0 0 -8233 -48723 -110280 = -7884500 0 0 0 0 0 0 0 0 0 0 0 0 151 0 0 0 0 -320 -1145 -1120 = -728000 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 0 0 216 -1209 -3170 = 491400 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 0 3069 15759 12780 = 14588600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 2459 8079 17440 = 11336000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 -344 -759 -5130 = -782600 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 78 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -143 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 221 Row-reduced echelon form of the extended system: 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 143 783 1180 = 767000 0 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -413 -2103 -2880 = -890500 0 0 15100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 381 1611 1560 = 1995500 0 0 0 3020 0 0 0 0 0 0 0 0 0 0 0 0 0 -19 -9 160 = 104000 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 0 0 -8 -123 -330 = -18200 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 0 3413 14253 20930 = 13604500 0 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 0 -3347 -6632 -8770 = -5700500 0 0 0 0 0 0 0 7550 0 0 0 0 0 0 0 0 0 7724 33819 50590 = 32883500 0 0 0 0 0 0 0 0 3020 0 0 0 0 0 0 0 0 1557 8367 12320 = 8008000 0 0 0 0 0 0 0 0 0 302 0 0 0 0 0 0 0 -24 -67 -84 = -54600 0 0 0 0 0 0 0 0 0 0 3775 0 0 0 0 0 0 -4812 -34347 -43720 = -12714000 0 0 0 0 0 0 0 0 0 0 0 15100 0 0 0 0 0 -8233 -48723 -110280 = -7884500 0 0 0 0 0 0 0 0 0 0 0 0 151 0 0 0 0 -320 -1145 -1120 = -728000 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 0 0 216 -1209 -3170 = 491400 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 0 3069 15759 12780 = 14588600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 0 2459 8079 17440 = 11336000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1510 -344 -759 -5130 = -782600 Top bound on feasible weight distribution on planes: 30 75 99 64 26 Feasible weight distribution of planes 1: [25, 15, 65, 50, 1, ] Feasible weight distribution of planes 2: [0, 65, 65, 0, 26, ] The number of solutions of the equations from Lemma 3.1: 532 The number of solutions of the extended system:2 The system of equations from Lemma 3.1 (feasible weight distributions of points): 1 1 1 1 1 = 156 3 9 15 21 27 = 2418 3 36 105 210 351 = 20553 Top bound on feasible weight distribution on points (without Zero equations): 91 156 156 86 30 Row-reduced echelon form for the equations from Lemma 3.1: 1 0 0 1 3 = 91 0 1 0 -3 -8 = -195 0 0 1 3 6 = 260 The extended system (The equations from Lemma 3.1 and the Zero equations): 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 654 1018 = 896584 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -721 -3522 -5066 = -2999360 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 711 2718 2982 = 4923880 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 -227 -102 1090 = 1021800 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 -5 -402 -1042 = -20800 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 6641 25770 43998 = 37457160 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 -2657 -6424 -13792 = -11053120 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 18053 63078 120386 = 100660040 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 4909 40782 49050 = 45981000 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 -55 -641 -119 = -228800 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 1829 -25362 -2986 = 7608640 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 -25795 -77550 -203270 = -107307200 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 -5895 -27800 -26160 = -24523200 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 935 -15570 -47130 = 3889600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 47277 153162 215514 = 259588160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 32405 155274 280130 = 262602600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 -635 -5682 -41590 = -2641600 1 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 91 0 1 0 -3 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = -195 0 0 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 260 Row-reduced echelon form of the extended system: 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 654 1018 = 896584 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -721 -3522 -5066 = -2999360 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 711 2718 2982 = 4923880 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 -227 -102 1090 = 1021800 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 -5 -402 -1042 = -20800 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 6641 25770 43998 = 37457160 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 -2657 -6424 -13792 = -11053120 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 18053 63078 120386 = 100660040 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 4909 40782 49050 = 45981000 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 -55 -641 -119 = -228800 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 1829 -25362 -2986 = 7608640 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 -25795 -77550 -203270 = -107307200 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 -5895 -27800 -26160 = -24523200 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 935 -15570 -47130 = 3889600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 47277 153162 215514 = 259588160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 32405 155274 280130 = 262602600 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 -635 -5682 -41590 = -2641600 Top bound on feasible weight distribution on points: 26 63 100 75 29 Feasible weight distribution of points 1: [26, 0, 65, 65, 0, ] Solving the equations from Lemma 3.2: Feasible weight distribution of planes 2: [0, 65, 65, 0, 26, ] Feasible weight distribution of points 1: [26, 0, 65, 65, 0, ] 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2061 -5390 -12090 9753432 | 1179672 0 7562 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 849 1714 4518 -3531840 | 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1698 3428 9036 -7063680 | 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5211 -1542 -33342 21677760 | 0 0 0 0 0 796 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -27 58 -214 112320 | 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -21 -176 917 87360 | 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -739 -432 -4280 3074240 | 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 372 417 2661 -1547520 | 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 42 352 1947 -174720 | 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6641 -25770 -43998 37457160 | 9830600 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2061 -5390 -12090 9753432 | 1179672 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1698 3428 9036 -7063680 | 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -6641 -25770 -43998 37457160 | 9830600 0 0 0 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -739 -432 -4280 3074240 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 42 352 1947 -174720 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -21 -176 917 87360 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 849 1714 4518 -3531840 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 796 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -27 58 -214 112320 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5211 -1542 -33342 21677760 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 372 417 2661 -1547520 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 -402 -1042 20800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -227 -102 1090 -1021800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 711 2718 2982 -4923880 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -721 -3522 -5066 2999360 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 654 1018 -896584 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -635 -5682 -41590 2641600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 935 -15570 -47130 -3889600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -55 -641 -119 228800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32405 155274 280130 -262602600 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5895 -27800 -26160 24523200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4909 40782 49050 -45981000 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47277 153162 215514 -259588160 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -25795 -77550 -203270 107307200 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18053 63078 120386 -100660040 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7562 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1829 -25362 -2986 -7608640 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2657 -6424 -13792 11053120 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6641 25770 43998 -37457160 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 654 1018 -896584 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -721 -3522 -5066 2999360 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 0 711 2718 2982 -4923880 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 0 -227 -102 1090 -1021800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30248 0 0 0 0 0 0 0 0 0 0 0 0 -5 -402 -1042 20800 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15124 0 0 0 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