2023/01/31 by Whan-Hyuk Choi, Hyun Jin Kim, Yoonjin Lee · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Error Correcting Code Techniques
paper · 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 Fq of length mℓ for every positive even integer ℓ. In this paper, we study the case where xm−1 has an arbitrary number of irreducible factors in Fq[x]; in the previous studies, only some special cases where xm−1 has exactly two or three irreducible factors in Fq[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 Fq of length mℓ for any even positive integer ℓ, where we require that q≡1(mod4) if the index ℓ≥6. By implementation of our method, we obtain a new optimal binary self-dual code [172,86,24], which is also a quasi-cyclic code of index 4.