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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, $v$ isan element in the group matrix ring $M_2(R)G$ and where $R$ is a finitecommutative Frobenius ring and $G$ is a finite group of order 18. We employthese generator matrices and search for binary $[72,36,12]$ self-dual codesdirectly over the finite field $\mathbb{F}_2.$ As a result, we find 134 Type Iand 1 Type II codes of this length, with parameters in their weight enumeratorsthat were not known in the literature before. We tabulate all of our findings.

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