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

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