Several classes of Galois self-orthogonal MDS codes and related applications

Let q = ph 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 = h2 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 selforthogonal (extended) GRS codes with odd q and 2e |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 eGalois 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 & RADIC;q 2 +1 are obtained.& COPY; 2023 Elsevier Inc. All rights reserved.