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We consider extended 1-perfect codes in Hamming graphs H (n, q). Such nontrivial codes are known only when n=2ᵏ, k 1, q=2, or n=q+2, q=2ᵐ, m 1. Recently, Bespalov proved nonexistence of extended 1-perfect codes for q=3, 4, n>q+2. In this work, we characterize all positive integers n, r and prime p, for which there exist such a code in H (n, pʳ).
Konstantin Vorob’ev (Sat,) studied this question.