Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Both self-dual codes and maximum distance separable (MDS) codes have nice algebraic structures, theoretical significance, and practical implications. We present two classes of the Hermitian self-dual, MDS, and generalized Reed–Solomon (GRS) codes. Conversely, we prove that the Hermitian self-dual, MDS, and GRS codes must be above two classes for
${n =4}$
and conjecture that this result also holds for each even length
$ {n}$
,
$4\le {n}\le \text {q+1}$
.