On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes

Hai Q. Dinh, Tushar Bag, Kanat Abdukhalikov, Sachin Pathak, Ashish K. Upadhyay, Ramakrishna Bandi, Warattaya Chinnakum

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet FqR, where R= Fq+ uFq+ vFq+ uvFq, with u2 = 1,v2 = 1, uv = vu and q = pm for odd prime p, positive integer m. Let (Formula presented.) be an automorphism on Fq. Extending (Formula presented.) to (Formula presented.) over R, we study skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR, where λ and Γ are units in Fq and R, respectively. We also show that, the dual of a skew (Formula presented.)-(λ,Γ)-constacyclic code over FqR is a skew (Formula presented.)-(λ− 1− 1)-constacyclic code over FqR. We classify some self-dual skew (Formula presented.)-(λ,Γ)-constacyclic codes using the possible values of units of R. Also using suitable values of λ,(Formula presented.),Γ and (Formula presented.), we present the structure of other linear codes over FqR. We construct a Gray map over FqR and study the Gray images of skew (Formula presented.)-(λ,Γ)-constacyclic codes over FqR. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew (Formula presented.)-cyclic codes than from cyclic codes over Fq.

Original languageEnglish
Pages (from-to)171-198
Number of pages28
JournalCryptography and Communications
Volume15
Issue number1
DOIs
Publication statusPublished - Feb 2023

Keywords

  • Gray map
  • Mixed alphabet codes
  • Optimal codes
  • Skew (Formula presented.) − (λ,Γ)-constacyclic codes

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Computational Theory and Mathematics
  • Applied Mathematics

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