On the Existence of Certain Optimal Self-Dual Codes with Lengths Between $74$ and $116$

TL;DR

This paper investigates the existence and construction of optimal binary self-dual codes with lengths between 74 and 116, introducing new codes and restrictions on their weight enumerators.

Contribution

It presents new self-dual codes of lengths 78 and 116, and establishes restrictions on weight enumerators for certain lengths, advancing understanding of optimal self-dual codes.

Findings

16 inequivalent self-dual [78,39,14] codes found, 4 with new weight enumerators.

At least 141 inequivalent self-dual [116,58,18] codes identified, mostly new.

Restrictions on weight enumerators exclude some possibilities for lengths 74, 76, 82, 98, 100.

Abstract

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.