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 Mk(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring Mk(R) are one sided ideals in the group matrix ring Mk(R)G and the corresponding codes over the ring R are Gk-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.
@article{arxiv.2102.00475,
title = {Group Matrix Ring Codes and Constructions of Self-Dual Codes},
author = {Steven Dougherty and Adrian Korban and Serap Sahinkaya and Deniz Ustun},
journal= {arXiv preprint arXiv:2102.00475},
year = {2021}
}