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).
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Konstantin Vorob’ev (2024) studied this question.
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