# Deep Holes of Projective Reed-Solomon Codes

## TL;DR

This paper explicitly constructs and classifies all deep holes in Projective Reed-Solomon codes with redundancy up to four, advancing understanding of their error structures using algebraic methods.

## Contribution

It introduces three classes of deep holes for PRS codes and completely classifies all deep holes with redundancy at most four, extending prior classifications.

## Key findings

- Three classes of deep holes explicitly constructed
- Complete classification for PRS codes with redundancy ≤ 4
- Advancement over previous work limited to redundancy ≤ 3

## Abstract

Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes --received words with maximum possible error distance-- for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy at most four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three in work arXiv:1612.05447