Multifold 1-perfect codes
Denis S. Krotov (Sobolev Institute of Mathematics)

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 -perfect code (-perfect code for list decoding) in any graph is a set of vertices such that every vertex of the graph is at distance not more than from exactly elements of . In -ary Hamming graphs, where is a prime power, we characterize all parameters of multifold -perfect codes and all parameters of additive multifold -perfect codes. In particular, we show that additive multifold -perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold -perfect codes always exist. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, additive codes, spreads, multispreads, completely regular codes, intriguing sets.
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Taxonomy
Topicsgraph theory and CDMA systems · Coding theory and cryptography
