$m$-adic residue codes over ${\mathbb{F}}_q[v]/(v^s-v)$ and their application to quantum codes

Abstract

Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{\mathbb{F}}_q\left[v\right]}{⟨v^s-v⟩}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{\mathbb{F}}_q\left[v\right]}{⟨v^s-v⟩}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{\mathbb{F}}_q\left[v\right]}{⟨v^s-v⟩}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{\mathbb{F}}_q\left[v\right]}{⟨v^s-v⟩}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{\mathbb{F}}_q\left[v\right]}{⟨v^s-v⟩}$ and give some examples to illustrate our findings.