MDS Codes with Euclidean and Hermitian Hulls of Flexible Dimensions and Their Applications to EAQECCs

TL;DR

This paper constructs new classes of MDS codes with flexible Euclidean and Hermitian hull dimensions using (extended) GRS codes, and applies them to develop new entanglement-assisted quantum error-correcting codes with improved parameters.

Contribution

It introduces novel MDS code classes with flexible hull dimensions and leverages them to create new EAQECCs, expanding the coding theory and quantum error correction fields.

Findings

Four new MDS codes with Hermitian hulls of flexible dimensions

Six new MDS codes with Euclidean hulls of flexible dimensions

New EAQECCs with length greater than q+1

Abstract

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of MDS codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. Specifically, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. For the former, we further construct four new classes of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new classes of MDS EAQECCs of length $n>q+1$. For the latter, we also give some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes.