🤖 AI Summary
This work addresses the loose lower bound on the Hermitian hull dimension of generalized rational algebraic geometry codes, which hinders the efficient construction of entanglement-assisted quantum error-correcting codes (EAQECCs). We propose a novel method integrating Weil differential residue properties with the Hermitian dual, enabling—for the first time—explicit lower-bound estimation and precise control of the Hermitian hull dimension. Leveraging algebraic function field theory and generalized Reed–Solomon code constructions, we systematically design two families of maximum-distance-separable (MDS) EAQECCs with unprecedented parameters: their lengths, dimensions, and required numbers of pre-shared ebits all surpass current records. The key innovation lies in establishing an analytic connection between the Hermitian hull dimension and the structure of Weil differentials, thereby providing a scalable algebraic-geometric framework for efficiently constructing high-performance EAQECCs.
📝 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.