Error-correcting codes have an important role in data storage and transmission and in cryptography, particularly in the post-quantum era. Hermitian matrices over finite fields and equipped with the rank metric have the potential to offer enhanced security with greater efficiency in encryption and decryption. One crucial tool for evaluating the error-correcting capabilities of a code is its weight distribution and the MacWilliams Theorem has long been used to identify this structure of new codes from their known duals. Earlier papers have developed the MacWilliams Theorem for certain classes of matrices in the form of a functional transformation, developed using $q$-algebra, character theory and Generalised Krawtchouk polynomials, which is easy to apply and also allows for moments of the weight distribution to be found. In this paper, recent work by Kai-Uwe Schmidt on the properties of codes based on Hermitian matrices such as bounds on their size and the eigenvalues of their association scheme is extended by introducing a negative-$q$ algebra to establish a MacWilliams Theorem in this form together with some of its associated moments. The similarities in this approach and in the paper for the Skew-Rank metric by Friedlander et al. have been emphasised to facilitate future generalisation to any translation scheme.

翻译：纠错码在数据存储、传输以及密码学中（特别是在后量子时代）扮演着重要角色。基于有限域且配备秩度量的厄米特矩阵，有望在加解密过程中以更高效率提供增强的安全性。评估纠错码纠错能力的关键工具之一是其重量分布，而MacWilliams定理长期以来被用于从已知对偶码的权重结构中识别新码的结构特征。早期文献已针对特定矩阵类建立了以函数变换形式呈现的MacWilliams定理，该定理通过$q$-代数、特征理论和广义Krawtchouk多项式推导，不仅易于应用，还可求取重量分布的矩。本文通过引入负$q$-代数构建此形式的MacWilliams定理及其相关矩，拓展了Kai-Uwe Schmidt近期关于基于厄米特矩阵的码性质（如大小界及其结合方案的谱）的研究工作。本文方法与Friedlander等人针对斜秩度量论文中的方法具有相似性，强调了这种相似性以便于未来向任意平移方案的推广。