Additive Polycyclic Codes over $\mathbb{F}_{4}$ Induced by Binary Vectors and Some Optimal Codes

TL;DR

This paper investigates the structure of additive polycyclic codes over 4 induced by binary vectors, providing generator polynomials, duals, and examples of codes with optimal or superior parameters for classical and quantum applications.

Contribution

It introduces a new class of additive polycyclic codes over 4, characterizes their structure, duals, and demonstrates their potential for constructing optimal and quantum codes.

Findings

Derived generator polynomials and cardinalities for these codes.

Established dual relationships, including Hermitian duals as sequential codes.

Presented examples of codes surpassing known optimal codes in size or performance.

Abstract

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $a$ in $\mathbb{F}_{2}^{n}.$ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $C$ is a right polycyclic code induced by a vector $a\in \mathbb{F}_{2}^{n}$, then the Hermitian dual of $C$ is a sequential code induced by $a.$ As an application of these codes, we present examples of additive right polycyclic codes over $\mathbb{F}_{4}$ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over $\mathbb{F}_{4}.$