On completely regular self-dual codes with covering radius $\rho \leq 3$

TL;DR

This paper classifies all self-dual completely regular codes with covering radius up to 3, identifying specific codes and families, and providing their intersection arrays.

Contribution

It offers a complete classification of self-dual completely regular codes with small covering radius, including new results for codes with radius 2 and 3.

Findings

Two sporadic codes of length 8 for  ho=2

An infinite family of length 4 for  ho=2

Only two codes with  ho=3: the extended ternary Golay code and a sum of Hamming codes

Abstract

We give a complete classification of self-dual completely regular codes with covering radius $\rho \leq 3$. For $\rho=1$ the results are almost trivial. For $\rho=2$, 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 $\rho=1$, each one. For $\rho=3$, in some cases, we use similar techniques to the ones used for $\rho=2$. 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 $\rho=3$ 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 $\rho=3$ is ternary and has length 12. We provide the intersection arrays for all such codes.