Structure of linear codes over the ring Bk

We study the structure of linear codes over the ring Bk which is defined by Fpr [v1, v2, . . . , vk]/〈v 2 i = vi, viv j = v jvi〉 k i, j=1. In order to study the codes, we begin with studying the structure of the ring Bk via a Gray map which also induces a relation between codes over Bk and codes over Fpr . We consider Euclidean and Hermitian self-dual codes, MacWilliams relations, as well as Singleton-type bounds for these codes. Further, we characterize cyclic and quasi-cyclic codes using their images under the Gray map, and give the generators for these type of codes.