Correctness and Corruption of Locally Decodable Codes

Locally decodable codes (LDCs) are error correcting codes with the extra property that it is sucient to read just a small number of positions of a possibly corrupted codeword in order to recover any one position of the input. To achieve this, it is necessary to use randomness in the decoding procedures. We refer to the probability of returning the correct answer as the correctness of the decoding algorithm. Thus far, the study of LDCs has focused on the question of the tradeo between their length and the query complexity of the decoders. Another natural question is what is the largest possible correctness, as a function of the amount of codeword corruption and the number of queries, regardless of the length of the codewords. Goldreich et al. (Computational Complexity 15(3), 2006) observed that for a given number of queries and fraction of errors, the correctness probability cannot be arbitrarily close to 1. However, the quantitative dependence between the largest possible correctness and the amount of corruption has not been established before. We present several bounds on the largest possible correctness for LDCs, as a function of the amount of corruption tolerated and the number of queries used, regardless of the length of the code. Our bounds are close to tight. We also investigate the relationship between the amount of corruption tolerated by an LDC and its minimum distance as an error correcting code. Even though intuitively the two notions are expected to be related, we demonstrate that in general this is not the case. However, we show a close relationship between minimum distance and amount of corruption tolerated for linear codes over arbitrary nite elds, and for binary nonlinear codes. We use these results to strengthen the known bounds on the largest possible amount of corruption that can be tolerated by LDCs (with any nontrivial correctness better than random guessing) regardless of the query complexity or the length of the code.

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