The Subfield Metric and its Application to Quantum Error Correction

TL;DR

A new weight and corresponding metric is introduced over finite extension fields for asymmetric error correction and the existence of an optimal family of codes achieving the Singleton-type upper bound is shown.

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.

Figures (3)

• Figure 1: Bounds on $A_{q^m,\lambda}(n,d)$ for $q=4,m=2,\lambda=4,d=7$, and $q=4,m=8,\lambda=5,d=10$.
• Figure 2: Bounds on $A_{q^m,\lambda}(n,d)$ for $q=521,m=4,\lambda=3,d=10$, and $q=5,m=12,\lambda=3,d=5$.
• Figure 3: Bounds on $A_{q^m,\lambda}(n,d)$ for $q=521,m=4,\lambda=3,d=10$, and $q=5,m=12,\lambda=3,d=5$.

Theorems & Definitions (46)

• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Definition 4
• Lemma 5
• proof
• Example 6
• Definition 7