Hermitian Hulls of Rational Algebraic Geometry Codes and Applications in Quantum Codes

TL;DR

This paper develops a method to determine Hermitian hull dimensions of algebraic geometry codes using algebraic function fields, enabling the construction of new entanglement-assisted quantum error-correcting codes with optimal parameters.

Contribution

It introduces a systematic approach to estimate Hermitian hull dimensions of AG codes and constructs new MDS EAQECCs with improved parameters.

Findings

Established a lower bound for Hermitian hull dimensions of AG codes.

Constructed two new families of MDS EAQECCs with optimal parameters.

Demonstrated the effectiveness of the approach using algebraic function fields.

Abstract

Interest in the hulls of linear codes has been growing rapidly. More is known when the inner product is Euclidean than Hermitian. A shift to the latter is gaining traction. The focus is on a code whose Hermitian hull dimension and dual distance can be systematically determined. Such a code can serve as an ingredient in designing the parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). We use tools from algebraic function fields of one variable to efficiently determine a good lower bound on the Hermitian hull dimensions of generalized rational algebraic geometry (AG) codes. We identify families of AG codes whose hull dimensions can be well estimated by a lower bound. Given such a code, the idea is to select a set of evaluation points for which the residues of the Weil differential associated with the Hermitian dual code has an easily verifiable property. The approach allows us to construct codes with designed Hermitian hull dimensions based on known results on Reed-Solomon codes and their generalization. Using the Hermitian method on these maximum distance separable (MDS) codes with designed hull dimensions yields two families of MDS EAQECCs. We confirm that the excellent parameters of the quantum codes from these families are new.