论文标题
MacWilliams的偏差等级度量标准
The MacWilliams Identity for the Skew Rank Metric
论文作者
论文摘要
错误校正代码的重量分布是确定其性能的关键统计量。将代码重量与双重重量相关联的一个关键工具是MacWilliams Identity，它首先是为锤量指标开发的。该身份具有两种形式：一种是重量枚举者的功能转换，而另一个是通过（广义）krawtchouk多项式的重量分布的直接关系。功能转换形式尤其可用于得出代码重量分布的重要力矩身份。在本文中，我们专注于偏斜等级指标中的代码。在这些代码中，代码字是偏斜的矩阵，两个矩阵之间的距离是偏斜等级度量标准，这是其差异的一半。本文以基于其相关偏斜等级指标下的偏斜对称矩阵的代码进行功能转换的形式开发了$ q $ -Analog MacWilliams身份。该方法引入了一个偏斜的Q $代数，并使用广义的krawtchouk多项式。基于这种新的MacWilliams身份，我们随后得出了这些代码的偏斜等级分布的一些时刻。
The weight distribution of an error correcting code is a crucial statistic in determining it's performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $q$-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$q$ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.
