In this paper, we show that, algebraically, the intersection of two GRS codes is a direct sum of some like-generalized Reed-Solomon codes, and that the dimension of such code can be given via the dimensions of the GRS codes and the degrees of some relevant polynomials. We also provide a necessary and sufficient condition for this intersection to be a GRS code. Our results naturally extend the main results in [11, 16, 19, 23]. Particularly, we deterministically construct two GRS codes with given code length, dimensions, and intersection dimension. As an application of our main results, we derive the algebraic structure of the hull of a GRS code and exhibit a necessary and sufficient condition for the hull to be a GRS code. In addition, we discuss when a GRS code is self-orthogonal or dual-containing and when the hull of an RS code is again an RS code. Finally, as an application, we resolve the problem of explicit construction of MDS EAQECCs from classical codes forn≤q. Several examples are included to illustrate our results.

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes over$\mathbb {F}_{q^{m}}$are generalized Reed-Solomon codes (GRS codes) when$m=1$. In this paper, we give a necessary and sufficient condition for the Galois dual codes of Goppa codes when$m=1$to be Goppa codes with the same locator sets. Furthermore, we show that the Galois hulls of the above codes are still the Goppa codes and determine their Goppa polynomials and dimensions. In particular, we characterize Galois linear complementary dual (LCD), Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such codes. Moreover, we apply these results to EAQECCs.

In this paper, we study the optimal RS codes and GRS codes with respect to the half-Singleton bound and the strict half-Singleton bound, respectively. We first provide an improved explicit construction of optimal RS codes that meet the half-Singleton bound. This explicit construction can obtain optimal RS codes of longer lengths than previous studies. Then we focus on the optimal GRS codes with respect to the strict half-Singleton bound. We prove the existence of such codes and provide an explicit construction of such codes over much larger fields.