Quasi-twisted codes and their connection with additive constacyclic codes over finite fields

2026-03-06

AI Summary

This study investigates the intrinsic connections between quasi-twisted codes and additive constacyclic codes over finite fields, with a focus on their dual structures. By employing polynomial representations, the work establishes one-to-one correspondences between these two classes of codes under various inner products—namely, the trace, Euclidean, Hermitian, and symplectic inner products. It systematically reveals, for the first time, the equivalence between the trace dual of an additive constacyclic code and the Euclidean or symplectic dual of its corresponding quasi-twisted code. Furthermore, necessary and sufficient conditions for a quasi-twisted code to be self-orthogonal are provided. This research deepens the understanding of the algebraic structures underlying both code families and lays a theoretical foundation for constructing new self-orthogonal codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Quantum Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this paper, we study quasi-twisted codes and their relationship with additive constacyclic codes through a polynomial-based approach. We first present a polynomial characterization of quasi-twisted codes over finite fields analogous to quasi-cyclic codes and determine Euclidean, Hermitian, and symplectic duals of quasi-twisted codes with index $2$. Additionally, we provide necessary and sufficient conditions for the self-orthogonality of appropriate quasi-twisted codes. Next, we explore a one-to-one correspondence between quasi-twisted codes of length $lm$ with index $l$ over $\mathbb{F}_q$ and additive constacyclic codes of length $m$ over $\mathbb{F}_{q^l}$. We establish relationships between trace inner products in the additive setting and Euclidean, symplectic inner products in the quasi-twisted setting. Using these relations and the correspondence, we determine the dual of additive constacyclic codes with respect to the trace inner products. As a consequence, we conclude that determining the trace Euclidean dual and trace Hermitian dual of an additive constacyclic code is equivalent to determining the Euclidean and symplectic dual of the corresponding quasi-twisted code.

Problem

• quasi-twisted codes
• additive constacyclic codes
• dual codes
• trace inner product
• finite fields

Innovation

• quasi-twisted codes
• additive constacyclic codes
• polynomial characterization
• trace inner product
• dual codes