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.
@article{10_37236_2929,
author = {J\"urgen Bierbrauer and Klaus Metsch},
title = {A bound on permutation codes},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {3},
doi = {10.37236/2929},
zbl = {1295.05007},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2929/}
}
TY - JOUR
AU - Jürgen Bierbrauer
AU - Klaus Metsch
TI - A bound on permutation codes
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/2929/
DO - 10.37236/2929
ID - 10_37236_2929
ER -
%0 Journal Article
%A Jürgen Bierbrauer
%A Klaus Metsch
%T A bound on permutation codes
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/2929/
%R 10.37236/2929
%F 10_37236_2929
Jürgen Bierbrauer; Klaus Metsch. A bound on permutation codes. The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/2929
