Maximum Distance Separable (MDS) self-dual codes are of significant theoretical and practical importance. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Correspondingly there have been many research on constructions of Euclidean self-dual MDS codes by using GRS codes. However, the study on Hermitian self-dual GRS codes is relatively limited. Since Hermitian self-dual GRS codes do not exist for $n>q+1$, this paper is devoted to an investigation of GRS codes in the case where $n\le q+1$. First, we prove that when $n\leq q+1$, there are only two classes of Hermitian self-dual GRS codes, confirming the conjecture in [13] and providing its proof simultaneously. Second, we present two explicit construction methods. Thus, the existence and construction of Hermitian self-dual GRS codes are fully solved.

翻译：最大距离可分（MDS）自对偶码具有重要的理论价值与实际意义。广义Reed-Solomon（GRS）码是最重要的一类MDS码。相应地，已有大量研究基于GRS码构造欧几里得自对偶MDS码。然而，关于Hermitian自对偶GRS码的研究相对有限。由于当$n>q+1$时Hermitian自对偶GRS码不存在，本文致力于研究$n\le q+1$情形下的GRS码。首先，我们证明当$n\leq q+1$时，仅存在两类Hermitian自对偶GRS码，这既证实了文献[13]中的猜想，也同时给出了其证明。其次，我们提出了两种显式构造方法。至此，Hermitian自对偶GRS码的存在性与构造问题得到了完整解决。