m-ADIC RESIDUE CODES OVER THE RING F-q[v]/(v(s) - v) AND THEIR APPLICATIONS TO QUANTUM CODES


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Kuruz F., Sarı M., Köroğlu M. E.

QUANTUM INFORMATION & COMPUTATION, vol.22, no.5-6, pp.427-439, 2022 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 22 Issue: 5-6
  • Publication Date: 2022
  • Doi Number: 10.26421/qic22.5-6-4
  • Journal Name: QUANTUM INFORMATION & COMPUTATION
  • Journal Indexes: Science Citation Index Expanded, Scopus, zbMATH
  • Page Numbers: pp.427-439
  • Keywords: m-adic residue codes, Cyclic codes, Quadratic residue codes, CSS construction, Quantum codes, CYCLIC 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 F-q[v]/< v(s)-v >. We determine the idempotent generators of the m-adic residue codes over F-q[v]/< v(s)-v >. We obtain some parameters of optimal m-adic residue codes over F-q[v]/< v(s)-v > with respect to Griesmer bound for rings. Furthermore, we derive a condition for m-adic residue codes over F-q [v]/< 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 F-q[v]/< v(s)-v > and give some examples to illustrate our findings.