Construction of quasi-cyclic self-dual codes over finite fields

• Published in: Linear & multilinear algebra Vol. 72; no. 6; pp. 1017 - 1043
• Main Authors: Choi, Whan-Hyuk, Kim, Hyun Jin, Lee, Yoonjin
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 12.04.2024; Taylor & Francis Ltd
• Subjects: Binary codes; Codes; Fields (mathematics); finite field; Integers; Quasi-cyclic code; self-dual code; self-reciprocal polynomial
• ISSN: 0308-1087; 1563-5139
• DOI: 10.1080/03081087.2023.2172377

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , 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 $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 ( mod 4 ) if the index $ \ell \ge 6 $ ℓ ≥ 6 . By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4.