Revisiting the Reed-Muller Locally Correctable Codes

Abstract

Local codes are a special kind of error-correcting codes. Locally correctable codes (LCCs) are one type of local codes. LCCs can efficiently recover any coordinate of its corrupted encoding by probing only a few but not all fraction of the corrupted word. A q-ary LCC which encodes length k messages to length N codewords with relative distance Δ has three parameters: r, δ and ϵ. r is called query complexity recording the number of queries. δ is called tolerance fraction measuring the relative distance between encoding codewords and its corrupted codes which can be locally decoded. ϵ is called error probability showing the coordinate of its corrupted encoding fail to be recovered with probability at most ϵ. One fundamental problem in LCCs is to determine the trade-off among rate, distance and query complexity. But for a specific LCC, focus is on query complexity, tolerance fraction and error probability. Reed-Muller codes (RM codes) are the most presentative LCCs. In order to understand the "local" more clearly, we revisit local correctors for RM codes and analyze them in detail: 1)The decoding procedures; 2)The role of Reed-Solomon codes (RS codes) in decoding RM LCCs; 3)Other local correctors for RM codes. How parameters including r, δ and ϵ change in RM LCCs have been analyzed in different correctors. We believe this paper can help us understand local codes better and grasp the main soul of this research direction.