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

We prove that the blocklength n of a linear 3-query locally correctable code (LCC) L : F^k → Fⁿ with distance δmust be at least n ≥ 2^{ω((δ2 k/(|F|-1)2)1/8)}. In particular, the blocklength of a linear 3-query LCC with constant distance over any small field grows exponentially with k. This improves on the best prior lower bound of n ≥ ω(k³), 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 binary Reed-Muller codes, which achieve n ≤ 2^O(k1/2). Because there is a 3-query LDC with a strictly subexponential blocklength, as a corollary we obtain the first strong separation between q-query LCCs and LDCs for any constant q ≥ 3. Our proof is based on a new upgrade of the method of spectral refutations via Kikuchi matrices developed in recent works that reduces establishing (non-)existence of combinatorial objects to proving unsatisfiability of associated XOR instances. Our key conceptual idea is to apply this method with XOR instances obtained via long-chain derivations - a structured variant of low-width resolution for XOR formulas from proof complexity.