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.
翻译：误差纠正码的重量分布是确定其性能的关键统计量。将码的重量与其对偶的重量联系起来的一个关键工具是MacWilliams恒等式，这个恒等式最初是为Hamming度量开发的。这个恒等式有两种形式：一种是权值枚举器的函数转换，另一种是通过（广义）Krawtchouk多项式直接关联重量分布。特别是函数变换形式可以用来推导码的重量分布的重要矩恒等式。在本文中，我们着重研究基于Skew Rank度量的码。在这些码中，码词是Skew-symmetric矩阵，而两个矩阵之间的距离是Skew Rank度量，它是它们之间差的秩的一半。本文开发了一个q-模拟的MacWilliams恒等式，用于基于Skew-symmetric矩阵的码，这些码使用它们相关的Skew Rank度量。本方法引入了一个Skew-q代数，并使用广义Krawtchouk多项式。基于这个新的MacWilliams恒等式，我们推导了这些码的Skew Rank分布的几个矩。
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