Structure of linear codes over the ring $$B_k$$ B k

Abstract

We study the structure of linear codes over the ring $$B_k$$ which is defined by $${\mathbb {F}}_{p^r}[v_1,v_2,\ldots ,v_k]/\langle v_i^2=v_i,~v_iv_j=v_jv_i \rangle _{i,j=1}^k$$ . In order to study the codes, we begin with studying the structure of the ring $$B_k$$ via a Gray map which also induces a relation between codes over $$B_k$$ and codes over $${\mathbb {F}}_{p^r}$$ . 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.