Multifold 1-perfect codes

Abstract

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.