On self-dual completely regular codes with covering radius ρ ≤ 3
Borges, Joaquim (Universitat Autònoma de Barcelona. Departament d'Enginyeria de la Informació i de les Comunicacions)
Zinoviev, Victor (Russian Academy of Sciences. A.A. Kharkevich Institute for Problems of Information Transmission)

Date:	2025
Abstract:	We give a complete classification of self-dual completely regular codes with covering radius ρ ≤ 3. For ρ = 1 the results are almost trivial. For ρ = 2, by using properties of the more general class of uniformly packed codes in the wide sense, we show that there are two sporadic such codes, of length 8, and an infinite family, of length 4, apart from the direct sum of two self-dual completely regular codes with ρ = 1, each one. For ρ = 3, in some cases, we use similar techniques to the ones used for ρ = 2. However, for some other cases we use different methods, namely, the Pless power moments which allow to us to discard several possibilities. We show that there are only two self-dual completely regular codes with ρ = 3 and d ≥ 3, which are both ternary: the extended ternary Golay code and the direct sum of three ternary Hamming codes of length 4. Therefore, any self-dual completely regular code with d ≥ 3 and ρ = 3 is ternary and has length 12. We provide the intersection arrays for all such codes.
Grants:	Agencia Estatal de Investigación PID2022-137924NB-I00 Agencia Estatal de Investigación RED2022-134306-T Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00643
Note:	Altres ajuts: acords transformatius de la UAB
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Language:	Anglès
Document:	Article ; recerca ; Versió sotmesa a revisió
Subject:	Completely regular codes ; Covering radius ; Self-dual codes
Published in:	Finite fields and their applications, Vol. 105 (August 2025) , art. 102617, ISSN 1090-2465

DOI: 10.1016/j.ffa.2025.102617

Preprint 39 p, 459.8 KB

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