An Exponential Lower Bound for Linear 3-Query Locally Correctable Codes

Abstract: A locally correctable code (LCC) is an error correcting code where one can recover any symbol of the original codeword by querying only a small number of randomly chosen symbols from the received corrupted codeword. In this talk, we present a proof that the blocklength n of a linear 3-query LCC C: {0,1}^k -> {0,1}^n must be at least \exp(k^{1/8}), i.e., it grows exponentially with k. This improves on the prior best lower bound of k^3 shown in our recent work, which holds even for the weaker setting of 3-query locally decodable codes (LDCs), and comes close to matching the best-known construction of 3-query LCCs based on Reed—Muller codes, which achieve a blocklength of n = \exp(\sqrt{k}). Because there is a 3-query LDC with a strictly subexponential blocklength, as a corollary we obtain the first separation between q-query LCCs and LDCs for any constant q. We subsequently show that any “design 3-LCC”, an LCC with a parity check matrix that is a block design, must have blocklength at least 2^{(1-o(1)) sqrt{k}}. As Reed—Muller codes are design 3-LCCs with blocklength n = 2^{2 sqrt{2k}}, this is tight up to a factor of 2 sqrt{2} in the exponent, and as a corollary resolves the Hamada conjecture for 4-designs up to this constant factor. For nonlinear LCCs, we give a generalization of our approach that proves a superpolynomial lower bound of k^{log k}, improving on the prior best lower bound of k^3.