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

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.

24 pages. arXiv admin note: substantial text overlap with arXiv:2102.00475; text overlap with arXiv:2102.00474