Locally Updatable and Locally Decodable Codes

We introduce the notion of locally updatable and locally decodable codes (LULDCs). In addition to having low decode locality, such codes allow us to update a codeword (of a message) to a codeword of a different message, by rewriting just a few symbols. While, intuitively, updatability and error-correction seem to be contrasting goals, we show that for a suitable, yet meaningful, metric (which we call the Prefix Hamming metric), one can construct such codes. Informally, the Prefix Hamming metric allows the adversary to arbitrarily corrupt bits of the codeword subject to one constraint – he does not corrupt more than a δ fraction (for some constant δ) of the t “most-recently changed” bits of the codeword (for all 1 ≤ t ≤ n, where n is the length of the codeword).