Quasi-cyclic codes of index 2 and skew polynomial rings over finite fields

Abstract

Let θ be the Frobenius automorphism of the finite field Fql over its subfield Fq, Fql[Y;θ] the skew polynomial ring and Fql[Y;θ]/〈Yl−1〉 the quotient ring of Fql[Y;θ] modulo its ideal 〈Yl−1〉. We construct a specific Fq-algebra isomorphism from Fql[Y;θ]/〈Yl−1〉 onto the matrix ring Ml(Fq), and investigate factorizations of polynomials in Fq[X] over Fq and Fql when l is a prime integer. Then we present an algorithm to calculate monic factors of Xm−1 in Fq2[Y;θ]/〈Y2−1〉[X], and construct a class of quasi-cyclic codes of length 2m and index 2 over Fq from these monic factors by use of an Fq-algebra isomorphism from Fq2[Y;θ]/〈Y2−1〉[X] onto M2(Fq)[X].