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, pp. 80-86] showed that any such code C : {0, 1}n → Σm 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|)q/(q-1)) without assuming that the decoder is nonadaptive.