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

Abstract

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field F q of length mℓ for every positive even integer ℓ. In this paper, we study the case where x m − 1 has an arbitrary number of irreducible factors in F q [ x ] ; in the previous studies, only some special cases where x m − 1 has exactly two or three irreducible factors in 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 F q of length mℓ for any even positive integer ℓ, where we require that q ≡ 1 ( mod 4 ) 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.