On existence of perfect bitrades in Hamming graphs

Abstract

A pair (T0,T1) 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 T0 and T1 or none simultaneously. The volume of a perfect bitrade (T0,T1) is the size of T0. If C0 and C1 are distinct perfect codes with minimum distance 3 in G then (C0∖C1,C1∖C0) is a perfect bitrade. For any q≥3, r≥1 we construct perfect bitrades of volume (q!)r in the Hamming graph H(qr+1,q) and show that for r=1 their volume is minimum.