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

TL;DR

This paper explores the algebraic structure of $m$-adic residue codes over a specific quotient ring, determines their parameters, and constructs quantum error-correcting codes from their Gray images, advancing coding theory and quantum computing.

Contribution

It studies the structure and parameters of $m$-adic residue codes over a quotient ring and constructs quantum codes from their dual-containing Gray images.

Findings

Determined idempotent generators of $m$-adic residue codes.

Obtained parameters of optimal codes respecting Griesmer bound.

Constructed quantum codes from dual-containing $m$-adic residue 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 $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ 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 $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings.