Group matrix ring codes and constructions of self-dual codes

Abstract

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.