Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA (N, k, s, t)'s is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of the system of equations. A branch-and-cut algorithm developed by Margot [2002. Pruning by isomorphism in branch-and-cut. Math. Programming Ser. A 94, 71-90; 2003a. Exploiting orbits in symmetric ILP. Math. Programming Ser. B 98, 3-21; 2003b. Small covering designs by branch-and-cut. Math. Programming Ser. B 94, 207-220; 2007. Symmetric ILP: coloring and small integers. Discrete Optim., 4, 40-62] for solving integer programming problems with large symmetry groups is used to find all non-isomorphic OA(24, 7, 2, 2)'s, CIA(24, k, 2, 3)'s for 6 <= k <= 11, OA(32, k, 2, 3)'s for 6 <= k <= 11, OA(40, k, 2, 3)'s for 6 <= k <= 10, OA(48, k, 2, 3)'s for 6 <= k <= 8, OA(56, k, 2, 3)'s, OA(80, k, 2,4)'s, OA(112, k, 2,4)'s, for k = 6, 7, OA(64, k, 2, 4)'s, OA(96, k, 2,4)'s for k = 7, 8, and OA(144, k, 2, 4)'s for k = 8, 9. Further applications to classifying covering arrays with the minimum number of runs and packing arrays with the maximum number of runs are presented. (C) 2007 Elsevier B.V. All rights reserved.