Lower bounds for adaptive 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 (On the efficiency of local decoding procedures for error correcting codes, STOC 2000, 80–86) showed that any such code C : {0, 1}n → Σ 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 show m = Ω((n/ log |Σ|)) without assuming that the decoder is non-adaptive.