论文标题
通过两个子集构建MD的欧几里得自动偶联代码
Construction of MDS Euclidean Self-Dual Codes via Two Subsets
论文作者
论文摘要
$ q $ -ary MD的欧几里得自偶代码的参数完全取决于其长度，并且在近年来已经广泛研究了具有新长度的MDS Euclidean自动偶数代码。在本文中，我们进一步研究了通过广义的Reed-Solomon（GRS）代码及其扩展代码来构建MDS Euclidean自动二重要代码。我们构建的主要思想是选择合适的评估点，以使相应的（扩展）GRS代码是欧几里得自我偶尔。 首先，我们认为评估集由两个不相交的子集组成，其中一个基于痕量函数，另一个是基于子空间及其cosets的联合。然后构建了四个新的MD欧几里得自偶联代码。其次，我们给出一个简单但有用的引理，以确保可以将两个有限场相交子集的对称差异作为所需的评估集。基于这种引理，我们概括了我们的第一个结构，并提供了两个新的MD欧几里得自偶二重要代码。最后，通过使用两个具有非空交点的乘法亚组及其coset，我们提出了具有灵活参数的MDS Euclidean自动偶联代码的三个通用构造。明确构建了MD欧几里得自动偶联代码的几个新家庭。
The parameters of a $q$-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The main idea of our construction is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we consider the evaluation set consists of two disjoint subsets, one of which is based on the trace function, the other one is a union of a subspace and its cosets. Then four new families of MDS Euclidean self-dual codes are constructed. Secondly, we give a simple but useful lemma to ensure that the symmetric difference of two intersecting subsets of finite fields can be taken as the desired evaluation set. Based on this lemma, we generalize our first construction and provide two new families of MDS Euclidean self-dual codes. Finally, by using two multiplicative subgroups and their cosets which have nonempty intersection, we present three generic constructions of MDS Euclidean self-dual codes with flexible parameters. Several new families of MDS Euclidean self-dual codes are explicitly constructed.
