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Extremal quasi-cyclic self-dual codes over finite fields
- Title
- Extremal quasi-cyclic self-dual codes over finite fields
- Authors
- Kim H.J.; Lee Y.
- Ewha Authors
- 이윤진
- SCOPUS Author ID
- 이윤진
- Issue Date
- 2018
- Journal Title
- Finite Fields and their Applications
- ISSN
- 1071-5797
- Citation
- Finite Fields and their Applications vol. 52, pp. 301 - 318
- Keywords
- Extremal code; Finite field; Quasi-cyclic code; Self-dual code
- Publisher
- Academic Press Inc.
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- 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. © 2018 Elsevier Inc.
- DOI
- 10.1016/j.ffa.2018.04.013
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
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