A Bound on Permutation Codes

Abstract

Consider the symmetric group $S_n$ with the Hamming metric. A permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ 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 $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ 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.

A lower bound on permutation codes of distance $n-1$

“…On the other hand, it is straightforward to see, [9], that M (n, n − 1) = n(n − 1) implies existence of a full set of MOLS (equivalently a projective plane) of order n, so any nontrivial upper bound on permutation codes would have major impact on design theory and finite geometry. This connection is explored in more detail in [5]. Permutation codes are used in [17] for some recent MOLS constructions.…”