Multifold 1‐perfect codes

A multifold 1‐perfect code (1‐perfect code for list decoding) in any graph is a set C $C$ of vertices such that every vertex of the graph is at distance not more than 1 from exactly μ $\mu $ elements of C $C$ . In q $q$ ‐ary Hamming graphs, where q $q$ is a prime power, we characterize all parameter...

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Bibliographic Details
Published inJournal of combinatorial designs Vol. 32; no. 9; pp. 546 - 555
Main Author Krotov, Denis S.
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.09.2024
Subjects
additive codes
Apexes
completely regular codes
Decoding
Graph theory
intriguing sets
list‐decoding codes
multifold packing
multiple covering
multispreads
Parameters
perfect codes
spreads
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Summary:A multifold 1‐perfect code (1‐perfect code for list decoding) in any graph is a set C $C$ of vertices such that every vertex of the graph is at distance not more than 1 from exactly μ $\mu $ elements of C $C$ . In q $q$ ‐ary Hamming graphs, where q $q$ 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.
ISSN:1063-8539
1520-6610
DOI:10.1002/jcd.21947