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.

中文翻译：

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来自$ M_2（R）G $的新单双偶二进制[72,36,12]自对偶代码-组矩阵环

在这项工作中，我们介绍了许多形式为$ [I_ {2n} \ |的生成器矩阵。\ \ tau_k（v）]，$其中$ I_ {kn} $是$ kn \ times kn $单位矩阵，$ v $是组矩阵环$ M_2（R）G $中的一个元素，其中$ R $是有限可交换的Frobenius环，$ G $是阶数为18的有限组。我们使用这些生成器矩阵，并直接在有限域$ \ mathbb {F中搜索二进制$ [72,36,12] $自对偶代码} _2。$结果，我们发现了134个这种长度的I型和1型II码，它们的权重枚举器中的参数在以前的文献中是未知的。我们将所有调查结果制成表格。