### Abstract

We study self-dual codes over a factor ring R=F_{q}[X]/(X^{m}−1) of length ℓ, equivalently, ℓ-quasi-cyclic self-dual codes of length mℓ over a finite field F_{q}, provided that the polynomial X^{m}−1 has exactly three distinct irreducible factors in F_{q}[X], where F_{q} 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 F_{q} 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 language | English |
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Pages (from-to) | 301-318 |

Number of pages | 18 |

Journal | Finite Fields and Their Applications |

Volume | 52 |

DOIs | |

Publication status | Published - 2018 Jul 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Finite Fields and Their Applications*,

*52*, 301-318. https://doi.org/10.1016/j.ffa.2018.04.013

}

*Finite Fields and Their Applications*, vol. 52, pp. 301-318. https://doi.org/10.1016/j.ffa.2018.04.013

**Extremal quasi-cyclic self-dual codes over finite fields.** / Kim, Hyun Jin; Lee, Yoonjin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Extremal quasi-cyclic self-dual codes over finite fields

AU - Kim, Hyun Jin

AU - Lee, Yoonjin

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85046665416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046665416&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2018.04.013

DO - 10.1016/j.ffa.2018.04.013

M3 - Article

AN - SCOPUS:85046665416

VL - 52

SP - 301

EP - 318

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

ER -