On quasi-cyclic codes as a generalization of cyclic codes

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic codes. We show a one-to-one correspondence between ℓ-quasi-cyclic codes of length mℓ and left ideals of Mℓ(Fq)[X]/(Xm−1). Then, we generalize BCH codes and evaluation codes in this context. We study their parameters and establish a key equation. Finally, we present a new [189,11,125]F4 code beating the known minimum distance for fixed length and dimension. Many codes with good parameters beating best known ones have been found from this latter.