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

2 n

|

τ

2

( v ) ] , where

I

2 n

is the 2 n × 2 n 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

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.