On existence of perfect bitrades in Hamming graphs

TL;DR

This paper investigates the existence and construction of perfect bitrades in Hamming graphs, providing explicit constructions and minimal volume results for certain parameters.

Contribution

It introduces new constructions of perfect bitrades in Hamming graphs and establishes minimal volume cases for specific parameters.

Findings

Constructed perfect bitrades in Hamming graphs of volume (q!)^r

Proved minimal volume for the case r=1

Connected perfect codes with perfect bitrades

Abstract

A pair $(T_0,T_1)$ of disjoint sets of vertices of a graph $G$ is called a perfect bitrade in $G$ if any ball of radius 1 in $G$ contains exactly one vertex in $T_0$ and $T_1$ or none simultaneously. The volume of a perfect bitrade $(T_0,T_1)$ is the size of $T_0$. In particular, if $C_0$ and $C_1$ are distinct perfect codes with minimum distance $3$ in $G$ then $(C_0\setminus C_1,C_1\setminus C_0)$ is a perfect bitrade. For any $q\geq 3$, $r\geq 1$ we construct perfect bitrades in the Hamming graph $H(qr+1,q)$ of volume $(q!)^r$ and show that for $r=1$ their volume is minimum.