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.

翻译：纠错码在数据存储、传输以及密码学（尤其是后量子时代）中扮演着重要角色。有限域上配备秩度量的Hermitian矩阵有望在加解密过程中以更高效率提供更强的安全性。评估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通过已知对偶码的结构来识别新码的这一结构。早期论文针对特定类别的矩阵发展了MacWilliams定理的函数变换形式，该形式借助$q$-代数、特征理论及广义Krawtchouk多项式建立，不仅易于应用，还能求解重量分布的矩。本文在Kai-Uwe Schmidt近期关于Hermitian矩阵基码性质（如码的尺寸界限及其结合方案的谱特征）的研究基础上，通过引入负$q$-代数建立该形式的MacWilliams定理及其相关矩，从而拓展了上述工作。