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：纠错码的权重分布是衡量其性能的关键统计量。关联码与其对偶码权重的重要工具之一是MacWilliams恒等式，该恒等式最初针对汉明度量提出。该恒等式有两种形式：一种是对权重枚举函数进行函数变换，另一种是通过（广义）Krawtchouk多项式直接关联权重分布。其中函数变换形式可用于推导码权重分布的重要矩恒等式。本文聚焦于斜秩度量下的码。此类码的码字为斜对称矩阵，两矩阵间的距离定义为斜秩度量，即其秩之差的一半。本文针对基于斜对称矩阵及其关联斜秩度量的码，开发了形如函数变换的$q$-模拟MacWilliams恒等式。该方法引入了斜-$q$代数，并使用了广义Krawtchouk多项式。基于这一新MacWilliams恒等式，本文进一步推导了此类码的斜秩分布的若干矩。