Perfect codes in Doob graphs

Abstract

We study \(1\)-perfect codes in Doob graphs \(D(m,n)\). We show that such codes that are linear over the Galois ring \(\mathrm {GR}(4^2)\) exist if and only if there exist integers \(\gamma \ge 0\) and \(\delta >0\) such that \(n=(4^{\gamma +\delta }-1)/3\) and \(m=(4^{\gamma +2\delta }-4^{\gamma +\delta })/6\). 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 \(2m+n = (4^\mu - 1)/3\) and \(m\le (4^\mu -5\cdot 2^{\mu -1}+1)/9\), we construct \(1\)-perfect codes in \(D(m,n)\), which do not necessarily have a group structure.