The MacWilliams Identity is a well established theorem relating the weight enumerator of a code to the weight enumerator of its dual. The ability to use a known weight enumerator to generate the weight enumerator of another through a simple transform proved highly effective and efficient. An equivalent relation was also developed by Delsarte which linked the eigenvalues of any association scheme to the eigenvalues of it's dual association scheme but this was less practical to use in reality. A functional transform was developed for some specific association schemes including those based on the rank metric, the skew rank metric and Hermitian matrices. In this paper those results are unified into a single consistent theory applied to these "Krawtchouk association schemes" using a $b$-algebra. The derivatives formed using the $b$-algebra have also been applied to derive the moments of the weight distribution for any code within these association schemes.

翻译：MacWilliams恒等式是一个经典定理，建立了码的重量计数器与其对偶码重量计数器之间的关系。通过简单变换利用已知重量计数器生成另一重量计数器的能力被证明极为高效且有效。Delsarte也提出了一个等价关系，将任意结合方案的特征值与其对偶结合方案的特征值相关联，但该关系在实际应用中实用性较低。针对某些特定结合方案（包括基于秩度量、斜秩度量和Hermite矩阵的方案）已发展出函数变换。本文将这些结果统一到一个一致的理论框架中，利用$b$-代数应用于这些“Krawtchouk结合方案”。基于$b$-代数生成的导数还被用于推导这些结合方案内任意码的重量分布矩。