New singly and doubly even binary [72,36,12] self-dual codes from M2(R)G - group matrix rings

Abstract

In this work, we present a number of generator matrices of the form [I2n|τ2(v)], where I2n is the 2n×2n identity matrix, v is an element in the group matrix ring M2(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 F2. 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.