On Determining Deep Holes of Generalized Reed-Solomon Codes

TL;DR

This paper fully classifies deep holes in generalized Reed-Solomon codes over prime fields with certain parameters, using deep hole trees and Erdős-Heilbronn conjecture results.

Contribution

It provides a complete classification of deep holes for generalized Reed-Solomon codes under specified conditions, advancing understanding of their structure.

Findings

Complete classification of deep holes for RS_p(D,k) codes.

Utilization of deep hole trees and Erdős-Heilbronn conjecture techniques.

Results applicable when |D| > k ≥ (p-1)/2.

Abstract

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 codes is proved to be co-NP-complete. For the extended Reed-Solomon codes $RS_q(\F_q,k)$, a conjecture was made to classify deep holes by Cheng and Murray in 2007. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes $RS_p (D,k)$, where $p$ is a prime, $|D| > k \geqslant \frac{p-1}{2}$. Our techniques are built on the idea of deep hole trees, and several results concerning the Erd{\"o}s-Heilbronn conjecture.