Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA(N,k,s,t)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)OA(24,7,2,2)'s, OA(24,k,2,3)OA(24,k,2,3)'s for 6⩽k⩽116⩽k⩽11, OA(32,k,2,3)OA(32,k,2,3)'s for 6⩽k⩽116⩽k⩽11, OA(40,k,2,3)OA(40,k,2,3)'s for 6⩽k⩽106⩽k⩽10, OA(48,k,2,3)OA(48,k,2,3)'s for 6⩽k⩽86⩽k⩽8, OA(56,k,2,3)OA(56,k,2,3)'s, OA(80,k,2,4)OA(80,k,2,4)'s, OA(112,k,2,4)OA(112,k,2,4)'s, for k=6,7k=6,7, OA(64,k,2,4)OA(64,k,2,4)'s, OA(96,k,2,4)OA(96,k,2,4)'s for k=7,8k=7,8, and OA(144,k,2,4)OA(144,k,2,4)'s for k=8,9k=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.