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The MacWilliams Identity is a well established theorem relating the weightenumerator of a code to the weight enumerator of its dual. The ability to use aknown weight enumerator to generate the weight enumerator of another through asimple transform proved highly effective and efficient. An equivalent relationwas also developed by Delsarte which linked the eigenvalues of any associationscheme to the eigenvalues of it's dual association scheme but this was lesspractical to use in reality. A functional transform was developed for somespecific association schemes including those based on the rank metric, the skewrank metric and Hermitian matrices. In this paper those results are unifiedinto a single consistent theory applied to these "Krawtchouk associationschemes" using a $b$-algebra. The derivatives formed using the $b$-algebra havealso been applied to derive the moments of the weight distribution for any codewithin these association schemes.

The MacWilliams Identity for Krawtchouk Association Schemes