## 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 | |

State | Published - Jul 2018 |

### Bibliographical note

Funding Information:The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574).

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Extremal code
- Finite field
- Quasi-cyclic code
- Self-dual code