A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation
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A code C {0,1}^k →{0,1}^n is a q-locally decodable code (q-LDC) if one can recover any chosen bit b_i of the message b ∈{0,1}^k with good confidence by randomly querying the encoding x := C(b) on at most q coordinates. Existing constructions of 2-LDCs achieve n = exp(O(k)), and lower bounds show that this is in fact tight. However, when q = 3, far less is known: the best constructions achieve n = exp(k^o(1)), while the best known results only show a quadratic lower bound n ≥Ω̃(k^2) on the blocklength. In this paper, we prove a near-cubic lower bound of n ≥Ω̃(k^3) on the blocklength of 3-query LDCs. This improves on the best known prior works by a polynomial factor in k. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs developed in [GKM22, HKM23] and, in particular, relies on bounding the spectral norm of appropriate Kikuchi matrices.
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