# Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

## TL;DR

This paper constructs new classes of MDS codes with flexible hull dimensions using GRS codes and applies these to develop novel entanglement-assisted quantum error-correcting codes with adaptable parameters.

## Contribution

It introduces methods to determine hull dimensions of GRS-based MDS codes and applies these to create new, flexible MDS EAQECCs with various entanglement requirements.

## Key findings

- Constructed MDS codes with all possible hull dimensions.
- Developed new MDS EAQECCs with flexible entanglement parameters.
- Extended the class of q-ary MDS EAQECCs for lengths greater than q+1.

## Abstract

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q + 1 are also obtained.