Optimality of Reed–Muller codes for constant-query LCCs

Determine whether, for every fixed integer q ≥ 2, the binary Reed–Muller codes achieve asymptotically minimal block length among all binary q-query locally correctable codes; equivalently, prove that no binary q-query locally correctable code has block length asymptotically smaller than the Reed–Muller construction as a function of message length k.

Background

The paper surveys known constructions of q-query locally correctable codes (LCCs), highlighting that Reed–Muller codes yield binary q-LCCs with block length n ≤ 2^{O(k^{1/(q−1)})}. Despite decades of research, no binary construction with asymptotically smaller block length is known, motivating a widely discussed optimality conjecture.

The authors’ new lower bounds concern q = 3, delivering sharp bounds for design 3-LCCs and superpolynomial bounds for smooth nonlinear 3-LCCs, but they do not settle the general optimality question for arbitrary constant q. Hence, determining the global optimality of Reed–Muller codes remains an explicit conjecture.