New cubic self-dual codes of length 54, 60 and 66

TL;DR

This paper constructs new binary cubic self-dual codes of lengths 54, 60, and 66 using algebraic methods, significantly expanding known code families and proposing a conjecture on their maximality.

Contribution

It introduces new binary cubic self-dual codes of specific lengths, improving previous results with a systematic algebraic construction approach.

Findings

Constructed 1 new code of length 54

Constructed 6 new codes of length 60

Constructed 50 new codes of length 66

Abstract

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.