A Bound on Permutation Codes

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| 6 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 6 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.