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：纠错码在数据存储与传输及密码学中具有重要作用，尤其是在后量子时代。基于有限域上的厄米特矩阵并配备秩度量的编码方案，有望在加密与解密过程中以更高效率实现更强安全性。评估纠错码纠错能力的关键工具是其重量分布，而MacWilliams定理长期以来被用于从已知对偶码的结构中识别新码的分布特征。已有论文针对特定矩阵类别，通过$q$-代数、特征理论及广义Krawtchouk多项式，以函数变换形式发展了MacWilliams定理，该定理不仅易于应用，还可推导重量分布的矩。本文对Kai-Uwe Schmidt近期关于基于厄米特矩阵的编码性质（如规模边界及其结合方案特征值）的研究进行拓展，通过引入负$q$代数建立了具有该形式的MacWilliams定理及其相关矩。本文方法与Friedlander等人针对斜秩度量论文中的方法的相似性被重点强调，以促进未来向任意平移方案的推广。