MDS codes in the Doob graphs

TL;DR

This paper investigates the existence and characterization of MDS codes in Doob graphs, establishing nonexistence results for large parameters and providing complete classifications for smaller cases and maximum distance.

Contribution

It provides the first comprehensive analysis of MDS codes in Doob graphs, including nonexistence proofs and complete characterizations for various parameter ranges.

Findings

No MDS codes exist for 2m+n>6 with 2<d<2m+n

All MDS codes with d≥3 are characterized for 2m+n≤6

MDS codes with maximum distance d=2m+n are fully characterized

Abstract

The Doob graph $D(m,n)$, where $m>0$, is the direct product of $m$ copies of The Shrikhande graph and $n$ copies of the complete graph $K_4$ on $4$ vertices. The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. In this paper we consider MDS codes in Doob graphs with code distance $d \ge 3$. We prove that if $2m+n>6$ and $2<d<2m+n$, then there are no MDS codes with code distance $d$. We characterize all MDS codes with code distance $d \ge 3$ in Doob graphs $D(m,n)$ when $2m+n \le 6$. We characterize all MDS codes in $D(m,n)$ with code distance $d=2m+n$ for all values of $m$ and $n$.