Polynomial representation of additive cyclic codes and new quantum codes

TL;DR

This paper introduces a polynomial framework for additive cyclic codes over finite fields, enabling their classification and the construction of new high-performance quantum codes with record-breaking parameters.

Contribution

It provides a polynomial representation for additive cyclic codes, classifies various self-dual and self-orthogonal codes, and constructs ten new record-breaking binary quantum codes.

Findings

Polynomial representation for additive cyclic codes over _{p^2}

Classification of symplectic self-dual and self-orthogonal codes

Ten record-breaking binary quantum codes

Abstract

We give a polynomial representation for additive cyclic codes over $\mathbb{F}_{p^2}$. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $\mathbb{F}_p$. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $\mathbb{F}_{p^2}$. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $\mathbb{F}_{4}$.