CSS-like Constructions of Asymmetric Quantum Codes

TL;DR

This paper extends the CSS construction for asymmetric quantum codes by removing linearity restrictions and using various inner products, leading to improved code parameters for multiple finite fields.

Contribution

It introduces CSS-like constructions that generalize existing methods, enabling the creation of better asymmetric quantum codes with broader code families.

Findings

Constructed many optimal CSS-like AQCs for various q-values.

Removed linearity restrictions to improve code parameters.

Achieved better error-correcting capabilities than standard CSS codes.

Abstract

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of $q$-ary AQCs is extended by removing the $\F_{q}$-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure $q$-ary CSS-like codes for $q \in {2,3,4,5,7,8,9}$ up to reasonable lengths are found. In many instances, removing the $\F_{q}$-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.