Large Hermitian hull GRS codes of any given length

The construction of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes of many specific lengths and large dimensions has been an active topic. The construction of Euclidean self-dual GRS codes and twisted generalized Reed-Solomon (TGRS) codes attracts some attentions. In this paper, we construct GRS [n, k, n-k+1]_q^2 codes (thus MDS codes) over F_q^2 of the arbitrary length n satisfying n ≤ q^2+1 and any given distance d satisfying d=O(q^2) , such that the dimensions h ≤ k of its Hermitian hull is at least h=O(k) . This work is a natural extension of previous constructions of Hermitian self-orthogonal GRS codes of many specific lengths. Our method can be used to construct large Hermitian hull MDS TGRS codes of the length n|(q^2-1) .