Optimality of Reed–Muller codes for constant-q locally correctable codes

Prove the conjecture that Reed–Muller codes are optimal q-query locally correctable codes over the binary alphabet for every fixed constant q ≥ 2; specifically, establish that any binary q-query locally correctable code with message length k must have block length at least 2^{Ω(k^{1/(q−1)})}, matching the asymptotic block length achieved by Reed–Muller codes.

Background

Reed–Muller codes give binary q-query locally correctable codes (q-LCCs) with block length n ≤ 2^{O(k^{1/(q−1)})} for constant q. Despite decades of work, no binary q-LCC construction with strictly smaller block length is known.

This motivates a central conjecture asserting the optimality of Reed–Muller codes for constant-query local correction. The paper advances lower bounds for the special case q = 3 (including design 3-LCCs and smooth nonlinear settings), but the general optimality conjecture across all constant q remains unresolved.