MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

翻译：MDS自对偶码具有良好的代数结构，其参数完全由码长决定。近年来，构造具有新长度的MDS欧几里得自对偶码已成为编码理论中的一个重要课题。本文致力于通过广义Reed-Solomon（GRS）码及其扩展（EGRS）码构造新的MDS欧几里得自对偶码。我们构造方法的核心在于寻找有限域的合适子集作为评估集，以确保对应的（扩展）GRS码是欧几里得自对偶的。首先，我们提出了一种从多个相交子集中选取评估集的方法，并给出了一个定理以保证所选评估集满足所需条件。其次，基于该定理，我们分别利用范数函数、三个乘法子群的并集及其陪集构造了六类新的MDS欧几里得自对偶码。最后，在我们的构造中，可能存在的MDS欧几里得自对偶码的比例超过85\%，远高于先前报道的结果。