Classical and Quantum Evaluation Codes at the Trace Roots

TL;DR

This paper introduces a new class of evaluation linear codes based on trace roots, providing conditions for self-orthogonality, and constructs quantum and classical codes with record-breaking parameters over finite fields.

Contribution

It presents a novel construction of evaluation codes using trace roots, enabling the creation of improved quantum and classical codes with record parameters.

Findings

Constructed stabilizer quantum codes over multiple finite fields with record parameters.

Derived classical linear codes over  with record parameters.

Provided conditions for self-orthogonality of these codes.

Abstract

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature and that are records at [http://www.codetables.de] for the binary case. Moreover, we obtain several classical linear codes over the field $\mathbb{F}_4$ which are records at [http://www.codetables.de].