Additive perfect codes in Doob graphs

TL;DR

This paper constructs a new class of additive perfect codes in Doob graphs, proving sufficiency of known conditions and providing explicit examples of quasi-cyclic codes.

Contribution

It introduces a 3-parameter class of additive perfect codes in Doob graphs and confirms the sufficiency of existing conditions for their existence.

Findings

Constructed a 3-parameter class of additive perfect codes.

Proved that necessary conditions for additive 1-perfect codes are sufficient.

Presented explicit quasi-cyclic additive 1-perfect codes in specific Doob graphs.

Abstract

The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$, where $n'+n''=n$. A set of vertices of $D(m,n)$ is called an additive code if it forms a subgroup of this group. We construct a $3$-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive $1$-perfect codes in $D(m,n'+n'')$ are sufficient. Additionally, two quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and $D(2667,0+127)$.