# On the Local Correctabilities of Projective Reed-Muller Codes

## TL;DR

This paper demonstrates that projective Reed-Muller codes are locally correctable with low query complexity and are shorter than generalized Reed-Muller codes, offering efficient decoding in the low-query regime.

## Contribution

It introduces a perfectly smooth local decoder for PRM codes when the degree is at most q-1, and shows PRM codes can be shorter than GRM codes while maintaining key parameters.

## Key findings

- PRM codes are locally correctable with low query complexity.
- A perfectly smooth local decoder is constructed for PRM codes.
- PRM codes can be shorter than GRM codes with similar query and message lengths.

## Abstract

In this paper, we show that the projective Reed-Muller~(PRM) codes form a family of locally correctable codes~(LCC) in the regime of low query complexities. A PRM code is specified by the alphabet size $q$, the number of variables $m$, and the degree $d$. When $d\leq q-1$, we present a perfectly smooth local decoder to recover a symbol by accessing $\gamma\leq q$ symbols to the coordinates fall on a line. There are three major parameters considered in LCCs, namely the query complexity, the message length and the code length. This paper shows that PRM codes are shorter than generalized Reed-Muller~(GRM) codes in LCCs. Precisely, given a GRM code over a field of size $q$, there exists a class of shorter codes over a field of size $q-1$, while maintaining the same values on the query complexities and the message lengths.