Structure of linear codes over the ring $B_k$

Abstract

We study the structure of linear codes over the ring $B_k$ which is defined by ${\mathbb{F}}_{p^r}[v_1,v_2,\dots ,v_k]/⟨v_i^2=v_i,v_i{v}_j=v_j{v}_i{⟩}_{i,j=1}^k.$ 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.

18 pages, accepted for publication (Journal of Applied Mathematics and Computing)