Generator matrices of quasi-cyclic codes

Official URL: http://dx.doi.org/10.1016/j.disc.2026.115235

In this paper, we provide a complete solution to an open problem by applying the spectral method to the polynomial generator matrices of quasi-cyclic codes of arbitrary index, using the corresponding reduced Gröbner basis of the given quasi-cyclic code. More precisely, with this method, we construct a full generator matrix over the suitable extension field. We provide two explicit examples to illustrate the construction of a full generator matrix from the given polynomial generator matrix of the code in the cases of index 2 and index 3, respectively. Furthermore, we investigate the reduced Gröbner basis of the subclass of 2D-cyclic codes of index 2 and index 3, proving that they have polynomial generator matrices in diagonal form.