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

TL;DR

This paper investigates quasi-twisted codes over finite fields, establishing their polynomial characterization, duals, and a correspondence with additive constacyclic codes, thereby advancing understanding of their structure and duality properties.

Contribution

It introduces a polynomial-based framework for quasi-twisted codes, explores their duals, and establishes a novel correspondence with additive constacyclic codes over finite fields.

Findings

Polynomial characterization of quasi-twisted codes

Duals of quasi-twisted codes determined

Correspondence between quasi-twisted and additive constacyclic codes established

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.