Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

TL;DR

This paper introduces a modified bordered construction using λ-circulant matrices to generate new binary self-dual codes with previously unknown weight enumerators at lengths 54, 68, 82, and 94.

Contribution

It presents a novel modification of the bordered construction method for self-dual codes, enabling the creation of new codes with unique weight enumerators.

Findings

Constructed new self-dual codes at lengths 54, 68, 82, 94

Identified previously unknown weight enumerators for these codes

Demonstrated the effectiveness of the modified construction method

Abstract

In this work, we define a modification of a bordered construction for self-dual codes which utilises $\lambda$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.