Hermitian Self-Dual GRS and Extended GRS Codes

Self-dual MDS codes over finite fields are linear codes with important cryptographic and combinatorial applications. In this correspondence, a different approach is proposed to obtain Hermitian self-dual MDS codes from (extended) generalized Reed-Solomon (abbreviation to GRS and EGRS) codes. The necessary and sufficient conditions of a Hermitian self-dual (extended) GRS code are presented. With the help of this new method, we revise and improve the results in Niu and Yue
et al.
(IEEE Communications Letters, 23(5): 781-784, 2019). Furthermore, we suppose that there exist Hermitian self-dual (extend) GRS codes over
$\mathbb {F}_{q^{2}}$
if and only if their code length
$n\leq q+1$
.