Duality on group algebras over finite chain rings: applications to additive group codes

2025-08-10

AI Summary

This study addresses the structural characterization of additive group codes and the construction of orthogonal complementary pairs over finite chain rings. To resolve these challenges, we introduce the notion of “additive complementary pairs” and employ finite chain ring extensions, R-bimodule isomorphisms, and involutive anti-automorphisms to construct symmetric non-degenerate trace-Euclidean inner products. Leveraging module decomposition theory, we establish a deep connection between code structures and representations of group algebras. Our main contributions are: (i) the first systematic algebraic characterization of additive group codes over finite chain rings; (ii) a duality theorem for orthogonal complementary pairs; (iii) the revelation of orthogonal duality relations for two-sided complementary codes under anti-automorphisms; and (iv) a novel algebraic framework for designing linear codes over finite chain rings.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Given a finite group $G$ and an extension of finite chain rings $S|R$, one can consider the group rings $mathscr{S} = S[G]$ and $mathscr{R} = R[G]$. The group ring $mathscr{S}$ can be viewed as an $R$-bimodule, and any of its $R$-submodules naturally inherits an $R$-bimodule structure; in the framework of coding theory, these are called emph{additive group codes}, more precisely a (left) additive group code of is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps $G$ to the standard basis of $S^n$, where $n=|G|$. In the first part of the paper, the ring extension $S|R$ is studied, and several $R$-module isomorphisms are established for decomposing group rings, thereby providing a characterization of the structure of additive group codes. In the second part, we construct a symmetric, nondegenerate trace-Euclidean inner product on $mathscr{S}$. Two additive group codes $mathcal{C}$ and $mathcal{D}$ form an emph{additive complementary pair} (ACP) if $mathcal{C} + mathcal{D} = mathscr{S}$ and $mathcal{C} cap mathcal{D} = {0}$. For two-sided ACPs, we prove that the orthogonal complement of one code under the trace-Euclidean duality is precisely the image of the other under an involutive anti-automorphism of $mathscr{S}$, linking coding-theoretical ACPs with module orthogonal direct-sum decompositions, representation theory, and the structure of group algebras over finite chain rings.

Problem

• Characterizing additive group codes via ring extensions and isomorphisms
• Constructing trace-Euclidean inner products for symmetric duality
• Linking additive complementary pairs to module decompositions and anti-automorphisms

Innovation

• Decompose group rings using R-module isomorphisms
• Construct symmetric trace-Euclidean inner product
• Link ACPs with anti-automorphism and duality