New Lower Bounds for General Locally Decodable Codes

For any odd integer q > 1, we improve the lower bound for general q-query locally decodablecodes C : {0, 1}n → {0, 1}m from m = Ω(n/ logn)q+1q−1 to m = Ω( nq+1q−1) / logn. For example,for q = 3 we improve the previous bound from Ω(n/ log n) to Ω(n/ logn). For linear 3-querylocally decodable codes C : F → F, we improve the lower bound further to Ω(n/ log log n),and our bound holds for any (possibly infinite) field F. Previously, the best lower bound for thiscase was Ω(n/ log n), and held only for constant-sized F. We are not aware of any previousnon-trivial lower bounds for large F and q > 2 queries.Our proofs use a random restriction of the message, hypergraph arguments, a new reductionfrom a q-query code to a generalization of a 2-query code, and quantum arguments. For linearcodes our proofs are completely elementary. We work with random linear projections and useadditional structure in the hypercube. The idea of using a random restriction (or projection forlinear codes) is new in this context, and may be a powerful technique for future work. 1Electronic Colloquium on Computational Complexity, Report No. 6 (2007)