On self-dual completely regular codes with covering radius ρ ≤ 3

Abstract

We give a complete classification of self-dual completely regular codes with covering radius $\rho \le 3$. 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 $d\ge 3$, 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 $d\ge 3$ and $\rho =3$ is ternary and has length 12.

We provide the intersection arrays for all such codes.