MacWilliams Identities for $m$-tuple Weight Enumerators

TL;DR

This paper generalizes MacWilliams identities to $m$-tuple weight enumerators, extending classical code theory results and providing new theoretical insights and illustrations for these generalized enumerators.

Contribution

It proves a new generalization of MacWilliams identities for $m$-tuple weight enumerators, building on prior work and offering a broader theoretical framework.

Findings

Established a generalized MacWilliams identity for $m$-tuple weight enumerators

Extended previous theorems of Britz, Ray-Chaudhuri, and Siap

Provided illustrative examples of $m$-tuple weight enumerators

Abstract

Since MacWilliams proved the original identity relating the Hamming weight enumerator of a linear code to the weight enumerator of its dual code there have been many different generalizations, leading to the development of $m$-tuple support enumerators. We prove a generalization of theorems of Britz and of Ray-Chaudhuri and Siap, which build on earlier work of Kl{\o}ve, Shiromoto, Wan, and others. We then give illustrations of these $m$-tuple weight enumerators.