The MacWilliams Identity for Krawtchouk Association Schemes

TL;DR

This paper unifies the MacWilliams Identity and related eigenvalue relations for Krawtchouk association schemes using a $b$-algebra, enabling efficient analysis of code weight distributions.

Contribution

It develops a unified theoretical framework for Krawtchouk association schemes, connecting weight enumerators, eigenvalues, and moments through a $b$-algebra.

Findings

Unified theory for Krawtchouk association schemes.

Derived moments of weight distributions for codes.

Simplified transform relations for eigenvalues and weight enumerators.

Abstract

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.