Better lower bounds for locally decodable codes

An error-correcting code is said to be locally decodable if a randomized algorithm can recover any single bit of a message by reading only a small number of symbols of a possibly corrupted encoding of the message. Katz and Trevisan (2000) showed that any such code C: {0, 1} → Σ^m with a decoding algorithm that makes at most q probes must satisfy m = Ω((n/log |Σ|)^q(q-1)/). They assumed that the decoding algorithm is non-adaptive, and left open the question of proving similar bounds for adaptive decoders. We improve the results of Katz and Trevisan (2000) in two ways. First, we give a more direct proof of their result. Second, and this is our main result, we prove that m = Ω((n/log|Σ|)^q(q-1)/) even if the decoding algorithm is adaptive. An important ingredient of our proof is a randomized method for smoothing an adaptive decoding algorithm. The main technical tool we employ is the Second Moment Method