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 characterize 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：在任何图中，多重1-完美码（用于列表解码的1-完美码）是一个顶点集合$C$，使得图中每个顶点与$C$中恰好$\mu$个元素的距离不超过$1$。在$q$元汉明图中（其中$q$为素数幂），我们刻画了多重1-完美码的所有参数以及可加多重1-完美码的所有参数。特别地，我们证明可加多重1-完美码与某种特殊的多重集推广的仿射展开（multispreads）相关，并且对应于多重1-完美码参数的multispreads总是存在。关键词：完美码，多重填充，多重覆盖，列表解码码，可加码，仿射展开，multispreads，完全正则码，引人集。