New Singly and Doubly Even Binary [72,36,12] Self-Dual Codes from $M_2(R)G$ -- Group Matrix Rings

TL;DR

This paper constructs new binary self-dual codes of length 72 using generator matrices derived from group matrix rings over a finite Frobenius ring, discovering 135 previously unknown codes.

Contribution

It introduces a novel method for generating binary self-dual codes from group matrix rings and provides new codes with unique parameters.

Findings

Discovered 134 Type I and 1 Type II codes of length 72.

All codes have parameters in their weight enumerators not previously documented.

The method can generate a large number of new self-dual codes.

Abstract

In this work, we present a number of generator matrices of the form $[I_{2n} \ | \ \tau_k(v)],$ where $I_{kn}$ is the $kn \times kn$ identity matrix, $v$ is an element in the group matrix ring $M_2(R)G$ and where $R$ is a finite commutative Frobenius ring and $G$ is a finite group of order 18. We employ these generator matrices and search for binary $[72,36,12]$ self-dual codes directly over the finite field $\mathbb{F}_2.$ As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.