Multiple contractions of permutation arrays

Abstract

Given a permutation σ on n symbols {0, 1, . . . , n − 1} and an integer 1 ≤ m ≤ n − 1, the mth contraction of σ is the permutation σ CT m on n − m symbols obtained by deleting the symbols n − 1, n − 2, . . . , n − m from the cycle decomposition of σ. The Hamming distance hd(σ, τ ) between two permutations σ and τ is the number of symbols x such that σ(x) = τ (x), and the Hamming distance of a non-empty set of permutations is the least Hamming distance among all pairs of distinct elements of the set. In this paper we identify how repeated contractions affect the Hamming distance between two permutations, and use it to obtain new lower bounds for the maximum possible size of a set of permutations on q − 1 symbols, for certain prime powers q, having Hamming distance q − 5, and for a set of permutations on q − m symbols, for certain prime powers q and 3 ≤ m ≤ 9, having Hamming distance q − 1 − 2m.