On Determining Deep Holes of Generalized Reed–Solomon Codes

For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed–Solomon (GRS) codes is proved to be co-NP-complete by Guruswami and Vardy. For the extended Reed–Solomon codes $RS_{q}( \mathbb {F}_{q},k)$ , a conjecture was made to classify deep holes by Cheng and Murray. Since then efforts have been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for GRS codes $RS_{p} (D,k)$ , where $p$ is a prime, $|D| > k \geqslant ({p-1})/{2}$ . Our techniques are built on the idea of deep hole trees, and several results concerning the Erdos–Heilbronn conjecture.