Generator Matrices of Quasi-cyclic Codes over Extension Fields Obtained from Gröbner Basis

Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every quasi-cyclic code is the subfield code and the trace code of a quasi-cyclic code over an extension field. The latter is defined by a parity check matrix obtained from a spectral analysis of a reduced Gröbner basis of the former. Moreover, it is shown that the quasi-cyclic code over the extension field and the one under consideration have the same length, dimension and minimum Hamming distance. Furthermore, we show that under certain conditions it is possible to construct a generator matrix of the quasi-cyclic code over the extension field using similar techniques to construct its parity check matrix. We illustrate that this construction is attainable for some good quasi-cyclic low density parity check codes like the [155, 64, 20] binary Tanner code.