Permutation codes invariant under isometries

Abstract

The symmetric group $$S_n$$ S n on $$n$$ n letters is a metric space with respect to the Hamming distance. The corresponding isometry group is well known to be isomorphic to the wreath product $$S_n \wr S_2$$ S n ? S 2 . A subset of $$S_n$$ S n is called a permutation code or a permutation array, and the largest possible size of a permutation code with minimum Hamming distance $$d$$ d is denoted by $$M(n, d)$$ M ( n , d ) . Using exhaustive search by computer on sets of orbits of isometry subgroups $$U$$ U we are able to determine serveral new lower bounds for $$M(n,d)$$ M ( n , d ) for $$n \le 22$$ n ≤ 22 . The codes are given by the group $$U$$ U and representatives of the $$U$$ U -orbits.