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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.18088 |
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Table of Contents:
- We give a complete classification of self-dual completely regular codes with covering radius $ρ\leq 3$. 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 $d\geq 3$, 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 $d\geq 3$ and $ρ=3$ is ternary and has length 12. We provide the intersection arrays for all such codes.