Equitable [[2, 10], [6, 6]]-partitions of the 12-cube

Abstract

We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7).