New constructions of self-dual generalized reed-solomon codes

A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are then obtained. Moreover, by the Mobius action over finite fields, for any known self-dual GRS codes, we give a systematic way to construct new self-dual GRS codes with flexible evaluation points.