On existence of perfect bitrades in Hamming graphs

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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.

Original languageEnglish
Article number112128
Number of pages7
JournalDiscrete Mathematics
Issue number12
Publication statusPublished - Dec 2020


  • Alternating group
  • MDS code
  • One-error-correcting code
  • Perfect bitrade
  • Perfect code
  • Spherical bitrade


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