Explicit constructions of optimal linear codes with Hermitian hulls and their application to quantum codes

TL;DR

This paper introduces new methods for constructing Hermitian self-orthogonal codes with large dimensions, leading to optimal quantum codes with flexible Hermitian hull sizes, using algebraic curves and puncturing techniques.

Contribution

It presents novel constructions of Hermitian self-orthogonal codes with large dimensions and arbitrary hull sizes, enabling the development of new quantum error-correcting codes.

Findings

Constructed Hermitian self-orthogonal codes with large dimensions.

Derived MDS and almost MDS codes with large Hermitian hulls.

Provided new entanglement-assisted quantum codes with improved parameters.

Abstract

We prove that any Hermitian self-orthogonal $[n,k,d]_{q^2}$ code gives rise to an $[n,k,d]_{q^2}$ code with $\ell$ dimensional Hermitian hull for $0\le \ell \le k$. We present a new method to construct Hermitian self-orthogonal $[n,k]_{q^2}$ codes with large dimensions $k>\frac{n+q-1}{q+1}$. New families of Hermitian self-orthogonal codes with good parameters are obtained; more precisely those containing almost MDS codes. By applying a puncturing technique to Hermitian self-orthogonal codes, MDS $[n,k]_{q^2}$ linear codes with Hermitian hull having large dimensions $k>\frac{n+q-1}{q+1}$ are also derived. New families of MDS, almost MDS and optimal codes with arbitrary Hermitian hull dimensions are explicitly constructed from algebraic curves. As an application, we provide entanglement-assisted quantum error correcting codes with new parameters.