The Subfield Metric and its Application to Quantum Error Correction

TL;DR

This paper introduces a novel weight and metric for finite extension fields tailored for asymmetric quantum error correction, establishing theoretical bounds and demonstrating optimal code families.

Contribution

It develops a new weight and metric for finite extension fields, specifically designed for asymmetric quantum codes, with theoretical bounds and optimal code existence proofs.

Findings

Defined a new weight and metric for asymmetric error correction

Established bounds and asymptotic behavior of codes using this metric

Proved the existence of optimal code families achieving the Singleton bound

Abstract

We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.