## Abstract

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field (Formula presented.) of length (Formula presented.) for every positive even integer ℓ. In this paper, we study the case where (Formula presented.) has an arbitrary number of irreducible factors in (Formula presented.); in the previous studies, only some special cases where (Formula presented.) has exactly two or three irreducible factors in (Formula presented.), were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over (Formula presented.) of length (Formula presented.) for any even positive integer ℓ, where we require that (Formula presented.) if the index (Formula presented.). By implementation of our method, we obtain a new optimal binary self-dual code (Formula presented.), which is also a quasi-cyclic code of index 4.

Original language | English |
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Pages (from-to) | 1017-1043 |

Number of pages | 27 |

Journal | Linear and Multilinear Algebra |

Volume | 72 |

Issue number | 6 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

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## Keywords

- Quasi-cyclic code
- finite field
- self-dual code
- self-reciprocal polynomial