Self-dual codes with group actions and invariants

TL;DR

This paper generalizes dual codes over finite rings with group actions, introduces $G$-dual codes, and extends MacWilliams identities and Gleason-type theorems to these settings, linking code invariants to group symmetries.

Contribution

It develops a comprehensive framework for $G$-codes over finite rings, including duality, weight enumerators, and invariance properties under Clifford--Weil groups, expanding classical coding theory.

Findings

Generalized Hayden's theorem for dual codes over finite rings.

Extended MacWilliams identities for $G$-codes and $G$-full weight enumerators.

Proved invariance of $G$-full weight enumerators of $G$-self-dual codes under Clifford--Weil groups.

Abstract

In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi's MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford--Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford--Weil groups and span the invariant subspaces of these groups.