We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. 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 $H(n,p^r)$.

翻译：我们考虑汉明图 $H(n,q)$ 中的扩展 $1$-完美码。此类非平凡码仅在 $n=2^k$, $k\geq 1$, $q=2$ 或 $n=q+2$, $q=2^m$, $m\geq 1$ 时已知存在。近期，Bespalov 证明了当 $q=3$, $4$ 且 $n>q+2$ 时扩展 $1$-完美码不存在。本文刻画了所有使得 $H(n,p^r)$ 中存在此类码的正整数 $n$、$r$ 和素数 $p$。