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| ≤ 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 \u3c 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