$t$-Frobenius Negacyclic Codes

TL;DR

This paper introduces and classifies a new class of quantum negacyclic codes called $t$-Frobenius negacyclic codes over finite fields, providing construction criteria, extending BCH distance, and identifying new codes.

Contribution

It defines $t$-Frobenius negacyclic codes, provides a complete classification, and extends BCH distance to these codes, advancing quantum code theory.

Findings

Complete classification of $t$-Frobenius negacyclic codes.

Construction criteria based on cyclotomic ring ideals.

Identification of new codes using extended BCH distance.

Abstract

Let $\mathbb{F}_p$ denote the finite field of order $p$, where $p$ is an odd prime. We study certain quantum negacyclic codes over $\mathbb{F}_p$ which we call $t$-Frobenius negacyclic codes. We obtain a criterion for constructing such codes from certain subspaces of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ in terms of generating pairs of ideals of cyclotomic rings and we completely classify all linear $t$-Frobenius negacyclic codes. Further, the notion of BCH distance is extended for these codes and several new codes are identified.