A $\mu$-fold (multifold) $1$-perfect code 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 graph, where $q$ is a prime power, we characterise all parameters of multifold $1$-perfect codes and all parameters of additive multifold $1$-perfect codes. Additionally, a description of parameters of additive $4$-ary completely regular codes of covering radius~$4$ is given. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, spreads, multispreads.
翻译：在任何图表中,一元一元(多倍)一元完美代码是一套固定的脊椎美元,使图表的每个顶端距离不超过美元,与美元每元一元的纯值元素不相上下。在以美元为主要功率的$q$-ary Hamming图中,我们描述了多倍一元完美代码的所有参数和添加剂多倍一元全值代码的所有参数。此外,还描述了覆盖半径~4美元的添加剂4美元全常规代码的参数。关键词:完美代码、多倍包装、多倍覆盖、列表解码代码、扩散、多频谱。
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