A bound on permutation codes

Document Type

Article

Publication Date

1-1-2013

Abstract

Consider the symmetric group Sn with the Hamming metric. A permutation code on n symbols is a subset C⊆ Sn If C has minimum distance ≥ n - 1, then |C| ≤ n2 - n. Equality can be reached if and only if a projective plane of order n exists. Call C embeddable if it is contained in a permutation code of minimum distance n - 1 and cardinality n2 - n. Let δ = δ (C) = n2 n - |C| be the deficiency of the permutation code C ⊆ Sn of minimum distance ≥ n - 1. We prove that C is embeddable if either δ ≤ 2 or if (δ2 - 1)(δ+1)2 < 27(n+2)/16. The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

Publication Title

Electronic Journal of Combinatorics

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