Nonbinary Quantum Codes from Two-Point Divisors on Hermitian Curves

TL;DR

This paper constructs asymmetric quantum error-correcting codes using two-point divisors on Hermitian curves, demonstrating improved parameters over one-point code-based codes through theoretical analysis and numerical examples.

Contribution

It introduces new asymmetric quantum codes from two-point Hermitian codes, showing they outperform one-point code-based codes in certain parameter regimes.

Findings

Strict improvements in code parameters over all finite fields for certain distances.

Large dimension pure AQECCs with better error correction capabilities.

Numerical results confirm the theoretical gains for F16 and F64.

Abstract

Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum error-correction, the observed presence of asymmetry in some quantum channels led to the study of asymmetric quantum codes (AQECCs) where we no longer assume that the different types of errors are equiprobable. This paper considers quantum codes constructed from the two-point codes. In the asymmetric case, we show strict improvements over all possible finite fields for a range of designed distances. We produce large dimension pure AQECC and small dimension impure AQECC that have better parameters than AQECC from one-point codes. Numerical results for the Hermitian curves over F16 and F64 are used to illustrate the gain.