## Abstract

There is a one-to-one correspondence between ℓ-quasi-cyclic codes over a finite field F _{q} and linear codes over a ring R=F _{q}[Y]/(Y ^{m}-1). Using this correspondence, we prove that every ℓ-quasi-cyclic self-dual code of length mℓ over a finite field F _{q} can be obtained by the building-up construction, provided that char(F _{q})=2 or q=1(mod4), m is a prime p, and q is a primitive element of F _{p}. We determine possible weight enumerators of a binary ℓ-quasi-cyclic self-dual code of length pℓ (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ℓ-quasi-cyclic codes of length 3ℓ) optimal self-dual codes of lengths 30,36,42,48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m=5, we obtain a new 8-quasi-cyclic self-dual [40,20,12] code over F _{3} and a new 6-quasi-cyclic self-dual [30,15,10] code over F _{4}. When m=7, we find a new 4-quasi-cyclic self-dual [28,14,9] code over F _{4} and a new 6-quasi-cyclic self-dual [42,21,12] code over F _{4}.

Original language | English |
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Pages (from-to) | 613-633 |

Number of pages | 21 |

Journal | Finite Fields and their Applications |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

### Bibliographical note

Funding Information:E-mail addresses: sunghyu@kut.ac.kr (S. Han), jl.kim@louisville.edu (J.-L. Kim), hsllee@ewha.ac.kr (H. Lee), yoonjinl@ewha.ac.kr (Y. Lee). 1 S. Han was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), which is supported by the Ministry of Education, Science and Technology (2010-0007232). 2 Y. Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0003516).

## Keywords

- Building-up construction
- Cubic code
- Quasi-cyclic self-dual code
- Quintic code