New extremal binary self-dual codes from a modified four circulant construction

TL;DR

This paper introduces a modified four circulant construction for self-dual codes, producing new extremal binary codes of lengths 64, 66, and 68 with previously unknown weight enumerators.

Contribution

It presents a novel modified four circulant construction and its bordered version, leading to the discovery of new extremal binary self-dual codes with unique parameters.

Findings

3 new codes of length 64

15 new codes of length 66

22 new codes of length 68

Abstract

In this work, we propose a modified four circulant construction for self-dual codes and a bordered version of the construction using the properties of \lambda-circulant and \lambda-reverse circulant matrices. By using the constructions on $F_2$, we obtain new binary codes of lengths 64 and 68. We also apply the constructions to the ring $R_2$ and considering the $F_2$ and $R_1$-extensions, we obtain new singly-even extremal binary self-dual codes of lengths 66 and 68. More precisely, we find 3 new codes of length 64, 15 new codes of length 66 and 22 new codes of length 68. These codes all have weight enumerators with parameters that were not known to exist in the literature.