Factorisation in M'(Fq)[X]. Construction of quasi-cyclic codes

Abstract

International audienceQuasi-cyclic codes are viewed as codes cancelled by polynomials with matricial coefficients. This construction leads to the problem of factorisation of Xm -1 in M'(Fq)[X]. In this paper we deal with the general factorisation in M'(Fq)[X]. Then we give results on the roots and the factorisation of the particular polynomial Xm -1. These factorisations permit the construction of such quasi-cyclic codes. We show that in most cases, these codes meet best known bounds for minimum distances. We even found two new codes with parameters better than known [30; 22; 6]F4 and [29; 21; 6]F4