Multifold 1-perfect codes

TL;DR

This paper characterizes all parameters of multifold 1-perfect codes in q-ary Hamming graphs, linking additive codes to multiset spreads and proving the existence of multispreads for these parameters.

Contribution

It provides a complete characterization of multifold 1-perfect codes and establishes their connection to multiset spreads and multispreads, including existence results.

Findings

All parameters of multifold 1-perfect codes in q-ary Hamming graphs are characterized.

Additive multifold 1-perfect codes are related to multiset spreads and multispreads.

Multispreads corresponding to these codes always exist.

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.