On the Local Correctabilities of Projective Reed-Muller Codes

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 ≤ q−1, we present a perfectly smooth local decoder to recover a symbol by accessing γ ≤ 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.