A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs, where $q$ is a prime power, we characterise all parameters of multifold $1$-perfect codes and all parameters of additive multifold $1$-perfect codes. In particular, we show that additive multifold $1$-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold $1$-perfect codes always exist. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, additive codes, spreads, multispreads, completely regular codes, intriguing sets.

翻译：在任意图中，多重1-完美码（列表解码的1-完美码）是一个顶点集合C，使得图中每个顶点到C中恰好μ个元素的距离不超过1。在q元汉明图（其中q为素数幂）中，我们刻画了多重1-完美码的所有参数以及加性多重1-完美码的所有参数。特别地，我们证明加性多重1-完美码与扩散（spreads）的特殊多重集推广——多重扩散（multispreads）相关，且对应于多重1-完美码的参数的多重扩散总是存在。关键词：完美码，多重堆积，多重覆盖，列表解码码，加性码，扩散，多重扩散，完全正则码，有趣集。