A duality for nonabelian group codes

TL;DR

This work formulates a duality for nonabelian group codes by introducing a representation-based dual $\mathcal{R}(\mathcal{H})$ built from irreducible representations. It proves a representation-extended Greene's Theorem and two MacWilliams identities, linking the weight enumerator of $\mathcal{H}$ to that of $\mathcal{R}(\mathcal{H})$ via a Tutte-polynomial framework, with $q=|\Gamma|$ and $T_P$ the polymatroid Tutte polynomial. When $\Gamma$ is abelian, the results specialize to classical MacWilliams identities, and the framework connects group codes to matroid theory through Greene's theorem. The paper also discusses practical challenges, such as multiplicities of representations and the need for encoding/decoding algorithms, outlining directions for future work and potential applications.

Abstract

In 1962, Jesse MacWilliams published a set of formulas for linear and abelian group codes that among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight enumerator and complete weight enumerator of a code to those of its dual code. A similar set of MacWilliams identities has been proven to exist for many other types of codes. In 2013, Dougherty, Solé, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In this paper, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this notion of duality. When the group is abelian, our results are equivalent to existing formulas in the literature.

Theorems & Definitions (13)

• Definition 2.1
• Lemma 2.2
• Example 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3