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Group Matrix Ring Codes and Constructions of Self-Dual Codes

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.