The Intersection of Two Generalized Reed-Solomon Codes

Abstract

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 on the hulls of GRS codes from existing literature. 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 entanglement-assisted quantum error-correcting codes (EAQECCs) from classical codes for $n \leq q$ . Several examples are included to illustrate our results.