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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. [ABSTRACT FROM AUTHOR]

Quantum maximum distance separable (MDS) codes form a significant class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed–Solomon codes, we construct two new classes of q -ary quantum MDS codes, which have minimum distance greater than q 2. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Let q = p h be a prime power and e be an integer with 0 ≤ e ≤ h − 1. e -Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e = 0) and Hermitian self-orthogonal codes (e = h 2 and h is even). In this paper, we propose two general methods to construct e -Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e -Galois self-orthogonal (extended) GRS codes with odd q and 2 e | h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e ′ -Galois self-orthogonal maximum distance separable (MDS) codes for all possible e ′ satisfying 0 ≤ e ′ ≤ h − 1 , new e -Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e -Galois hull via the punctured codes are derived. Moreover, some new q -ary quantum MDS codes with length greater than q + 1 and minimum distance greater than q 2 + 1 are obtained. [ABSTRACT FROM AUTHOR]