Structure of linear codes over the ring $B_k$

TL;DR

This paper investigates the structure of linear codes over the ring B_k, exploring their properties, dualities, bounds, and classifications, and establishing connections with codes over finite fields through Gray maps.

Contribution

It provides a detailed structural analysis of codes over B_k, including self-duality, cyclicity, and generator characterization, extending coding theory over finite rings.

Findings

Characterization of Euclidean and Hermitian self-dual codes over B_k

Derivation of MacWilliams relations and Singleton bounds for these codes

Explicit description of cyclic and quasi-cyclic codes and their generators

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.