Perfect codes in Doob graphs

TL;DR

This paper characterizes the existence of 1-perfect codes in Doob graphs, providing conditions for linear and additive codes over specific rings, and proves the existence of such codes for certain parameters.

Contribution

It establishes necessary and sufficient conditions for the existence of 1-perfect codes in Doob graphs, including new existence results for codes without group structure restrictions.

Findings

Linear 1-perfect codes exist for specific parameters related to powers of 4.

Necessary conditions for additive codes over Z_4 are derived.

Existence of 1-perfect codes is proven for certain parameter ranges without group structure restrictions.

Abstract

We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \ge 0$ and $d>0$. We also prove necessary conditions on $(m,n)$ for $1$-perfect codes that are linear over $Z_4$ (we call such codes additive) to exist in $D(m,n)$ graphs; for some of these parameters, we show the existence of codes. For every $m$ and $n$ satisfying $2m+n=(4^t-1)/3$ and $m \le (4^t-5\cdot 2^{t-1}+1)/9$, we prove the existence of $1$-perfect codes in $D(m,n)$, without the restriction to admit some group structure. Keywords: perfect codes, Doob graphs, distance regular graphs.