On completely regular self-dual codes with covering radius $ρ\leq 3$
J. Borges, V. A. Zinoviev
TL;DR
This work delivers a complete classification of self-dual completely regular codes with covering radius up to $3$, combining UPWS and design-theoretic methods, direct-sum constructions, and Pless power moments. For $\rho=1$, the codes are essentially trivial; for $\rho=2$, the classification yields two sporadic length-$8$ examples and an infinite length-$4$ family beyond direct sums of $\rho=1$ codes; for $\rho=3$, only two ternary codes exist with $d\ge 3$—the extended ternary Golay code and the direct sum of three ternary Hamming codes—implying any such code with $d\ge 3$ and $\rho=3$ has length $12$ and is ternary. The paper also provides intersection arrays for all identified codes and rules out numerous parameter combinations via Pless moments and design-based arguments. Overall, it delivers a complete, explicit catalog of self-dual CR codes up to $\rho=3$, clarifying the role of field size and length in these highly structured codes. The results have implications for constructing small, highly regular codes and for understanding the limitations of CR-structure in low covering radii.
Abstract
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.