m-ADIC RESIDUE CODES OVER THE RING Fq [v]/(vs - v) AND THEIR APPLICATIONS TO QUANTUM CODES


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Kuruz F. , Sarı M. , Köroğlu M. E.

Quantum Information and Computation, vol.22, no.5-6, pp.427-439, 2022 (Journal Indexed in SCI Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 22 Issue: 5-6
  • Publication Date: 2022
  • Doi Number: 10.26421/qic22.5-6-4
  • Title of Journal : Quantum Information and Computation
  • Page Numbers: pp.427-439
  • Keywords: CSS construction, Cyclic codes, M-adic residue codes, Quadratic residue codes, 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 $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. 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.