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Highrate codes with sublineartime decoding
, 2010
"... Locally decodable codes are errorcorrecting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding al ..."
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Locally decodable codes are errorcorrecting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been well studied, and it has widely been suspected that nontrivial locality must come at the price of low rate. A particular setting of potential interest in practice is codes of constant rate. For such codes, decoding algorithms with locality O(k ɛ) were known only for codes of rate exp(1/ɛ), where k is the length of the message. Furthermore, for codes of rate> 1/2, no nontrivial locality has been achieved. In this paper we construct a new family of locally decodable codes that have very efficient local decoding algorithms, and at the same time have rate approaching 1. We show that for every ɛ> 0 and α> 0, for infinitely many k, there exists a code C which encodes messages of length k with rate 1 − α, and is locally decodable from a constant fraction of errors using O(k ɛ) queries and time. The high rate and local decodability are evident even in concrete settings (and not just in asymptotic behavior), giving hope that local decoding techniques may have practical implications. These codes, which we call multiplicity codes, are based on evaluating high degree multivariate polynomials and their derivatives. Multiplicity codes extend traditional multivariate polynomial based codes; they inherit the localdecodability of these codes, and at the same time achieve better tradeoffs and flexibility in their rate and distance.
Three query locally decodable codes with higher correctness require exponential length
 In Proceedings of STACS 2011
"... 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 resu ..."
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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 3query 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 − 3δ for nonbinary codes, where an adversary is allowed to corrupt up to δ fraction of the codeword. The largest correctness for a subexponential length 3query binary code is achieved in a construction by Woodruff, and it is below 1 − 3δ. We show that achieving slightly larger correctness (as a function of δ) requires exponential codeword length for 3query 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 3query linear codes. Our lower bounds hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for qquery 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.
Lower Bounds for Approximate LDCs
"... We study an approximate version of qquery LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A qquery (α, δ)approximate LDC is a set V of n points in Rd so that, for each i ∈ [d] there are Ω(δn) disjoint qtuples (u1,...,uq) in V so t ..."
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We study an approximate version of qquery LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A qquery (α, δ)approximate LDC is a set V of n points in Rd so that, for each i ∈ [d] there are Ω(δn) disjoint qtuples (u1,...,uq) in V so that span(u1,...,uq) contains a unit vector whose i’th coordinate is at least α. We prove exponential lower bounds of the form n ≥ 2Ω(αδ d) for the case q = 2 and, in some cases, stronger bounds (exponential in d). 1