Extremal quasi-cyclic self-dual codes over finite fields

Hyun Jin Kim, Yoonjin Lee

Research output: Contribution to journalArticle

Abstract

We study self-dual codes over a factor ring R=Fq[X]/(Xm−1) of length ℓ, equivalently, ℓ-quasi-cyclic self-dual codes of length mℓ over a finite field Fq, provided that the polynomial Xm−1 has exactly three distinct irreducible factors in Fq[X], where Fq is the finite field of order q. There are two types of the ring R depending on how the conjugation map acts on the minimal ideals of R. We show that every self-dual code over the ring R of the first type with length ≥6 has free rank ≥2. This implies that every ℓ-quasi-cyclic self-dual code of length mℓ≥6m over Fq can be obtained by the building-up construction, where m corresponds to the ring R of the first type. On the other hand, there exists a self-dual code of free rank ≤1 over the ring R of the second type. We explicitly determine the forms of generator matrices of all self-dual codes over R of free rank ≤1. For the case that m=7, we find 9828 binary 10-quasi-cyclic self-dual codes of length 70 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the second type. These codes are all new codes. Furthermore, for the case that m=17, we find 1566 binary 4-quasi-cyclic self-dual codes of length 68 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the first type.

Original languageEnglish
Pages (from-to)301-318
Number of pages18
JournalFinite Fields and Their Applications
Volume52
DOIs
Publication statusPublished - 2018 Jul 1

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Quasi-cyclic Codes
Self-dual Codes
Galois field
Polynomials
Ring
Equivalence
Binary
Conjugation
Generator

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Algebra and Number Theory
  • Engineering(all)
  • Applied Mathematics

Cite this

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title = "Extremal quasi-cyclic self-dual codes over finite fields",
abstract = "We study self-dual codes over a factor ring R=Fq[X]/(Xm−1) of length ℓ, equivalently, ℓ-quasi-cyclic self-dual codes of length mℓ over a finite field Fq, provided that the polynomial Xm−1 has exactly three distinct irreducible factors in Fq[X], where Fq is the finite field of order q. There are two types of the ring R depending on how the conjugation map acts on the minimal ideals of R. We show that every self-dual code over the ring R of the first type with length ≥6 has free rank ≥2. This implies that every ℓ-quasi-cyclic self-dual code of length mℓ≥6m over Fq can be obtained by the building-up construction, where m corresponds to the ring R of the first type. On the other hand, there exists a self-dual code of free rank ≤1 over the ring R of the second type. We explicitly determine the forms of generator matrices of all self-dual codes over R of free rank ≤1. For the case that m=7, we find 9828 binary 10-quasi-cyclic self-dual codes of length 70 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the second type. These codes are all new codes. Furthermore, for the case that m=17, we find 1566 binary 4-quasi-cyclic self-dual codes of length 68 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the first type.",
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Extremal quasi-cyclic self-dual codes over finite fields. / Kim, Hyun Jin; Lee, Yoonjin.

In: Finite Fields and Their Applications, Vol. 52, 01.07.2018, p. 301-318.

Research output: Contribution to journalArticle

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