On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs

TL;DR

This paper explores linear codes with one-dimensional Euclidean hulls, constructing optimal codes from algebraic geometry and applying them to develop new entanglement-assisted quantum error-correcting codes (EAQECCs).

Contribution

It introduces new classes of optimal linear codes with one-dimensional Euclidean hulls derived from algebraic geometry, and demonstrates their application to EAQECCs.

Findings

Constructed several classes of optimal linear codes with one-dimensional Euclidean hull.

Presented new entanglement-assisted quantum error-correcting codes (EAQECCs).

Highlighted the importance of hull dimension in code automorphism and equivalence algorithms.

Abstract

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp$. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes. Several classes of optimal linear codes with one-dimensional Euclidean hull are constructed. Some new EAQECCs are presented.