On existence of perfect bitrades in Hamming graphs
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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∖ C_1,C_1∖ C_0) is a perfect bitrade. For any q≥ 3, r≥ 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.
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