论文标题
关于自二重广义的芦苇 - 固体代码的注释
A Note on Self-Dual Generalized Reed-Solomon Codes
论文作者
论文摘要
如果与欧几里得内部产品相对于MDS代码，则称为MDS自偶代码。此类代码的参数完全由代码长度确定。在本文中，我们通过通用的Reed-Solomon（GRS）代码及其扩展代码考虑了MDS自动二倍代码的新结构。我们构造的关键思想是选择合适的评估点，以便相应的（扩展）GRS代码是自动划分的。我们构造的评估集由一个有限字段及其cosets组成，在更大的子组中。获得了四个新的MDS自偶代码系列，它们的参数比某些区域的以前结果更好。此外，通过对有限领域的Mobius动作，我们提供了一种系统的方式，可以提供不同评估点的自动划分GRS代码，提供了任何已知的自动划分GRS代码。特别是，我们证明，$ \ mathbb {f} _ {q} $带有长度$ n <q+1 $的所有自duald扩展GRS代码可以从具有相同参数的GRS代码构建。
A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over $\mathbb{F}_{q}$ with length $n< q+1$ can be constructed from GRS codes with the same parameters.
