On MDS Codes With Galois Hulls of Arbitrary Dimensions

TL;DR

This paper explores the properties of Galois hulls in linear codes, specifically (extended) GRS codes, and introduces new methods to construct MDS codes with Galois hulls of any dimension, expanding the known classes of such codes.

Contribution

It presents novel constructions of MDS codes with Galois hulls of arbitrary dimensions using (extended) GRS codes and introduces two general construction methods.

Findings

New classes of MDS codes with Galois hulls of arbitrary dimensions are constructed.

Methods based on Hermitian or Galois self-orthogonal (extended) GRS codes are effective.

Relatively strict conditions enable the creation of codes with larger Galois hulls.

Abstract

The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict conditions can also lead to many new classes of MDS codes with Galois hulls of arbitrary dimensions.