Three Query Locally Decodable Codes with Higher Correctness Require Exponential Length
Résumé
Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding, possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is $1 - 3 \delta$ for nonbinary codes, where an adversary is allowed to corrupt up to $\delta$ fraction of the codeword. The largest correctness for a subexponential length $3$-query binary code is achieved in a construction by Woodruff, and it is below $1 - 3 \delta$. We show that achieving slightly larger correctness (as a function of $\delta$) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of $3$-query linear codes. Our results hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for q-query codes for $q>3$, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms. In addition, our results imply tradeoffs on the parameters of error correcting data structures.
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