In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring $M_k(R)$ and the ring $R,$ where $R$ is the commutative Frobenius ring. We show that codes over the ring $M_k(R)$ are one sided ideals in the group matrix ring $M_k(R)G$ and the corresponding codes over the ring $R$ are $G^k$-codes of length $kn.$ Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters $[72,36,12]$ and find new singly-even and doubly-even codes of this type. In particular, we construct $16$ new Type~I and $4$ new Type~II binary $[72,36,12]$ self-dual codes.

翻译：在这项工作中,我们研究了由来自集团矩阵环的元素产生的代码。我们提出了一个矩阵构造,我们用来在两个不同的环境空间生成代码:矩阵环$M_k(R)美元和环R美元,其中R美元是通性Frobenius环。我们显示,美元M_k(R)美元上的代码是集团矩阵的单向理想之一,环上相应的代码是美元M_k(R)G美元,而环上的相应代码是美元长度为$G$-k$的代码。此外,我们提供了由上述矩阵构造构成的自体代码的生成矩阵矩阵。我们使用这一生成矩阵来寻找带有参数($72,36,12美元)的二元自体代码,并找到这种类型的新的单数和双倍代码。特别是,我们建造了16美元的新类型~I和4美元新的类型~二($47,36,12美元)的自体代码。