Xing-Ling Codes, Duals of their Subcodes, and Good Asymmetric Quantum Codes

TL;DR

This paper constructs high-quality asymmetric quantum error-correcting codes using Xing-Ling classical codes, achieving optimal sizes for certain parameters and advancing quantum coding theory.

Contribution

It introduces a method to build asymmetric quantum codes from Xing-Ling codes, demonstrating their optimality in specific parameter ranges.

Findings

Constructed quantum codes correct multiple error types.

Achieved codes with sizes matching the best classical codes.

Applicable for certain code lengths and field sizes.

Abstract

A class of powerful $q$-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are $q$-ary block codes that encode $k$ qudits of quantum information into $n$ qudits and correct up to $\flr{(d_{x}-1)/2}$ bit-flip errors and up to $\flr{(d_{z}-1)/2}$ phase-flip errors.. In many cases where the length $(q^{2}-q)/2 \leq n \leq (q^{2}+q)/2$ and the field size $q$ are fixed and for chosen values of $d_{x} \in \{2,3,4,5\}$ and $d_{z} \ge \delta$, where $\delta$ is the designed distance of the Xing-Ling (XL) codes, the derived pure $q$-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.