## Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits (2007)

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Venue: | SIAM J. COMPUT |

Citations: | 30 - 9 self |

### BibTeX

@ARTICLE{Dvir07locallydecodable,

author = {Zeev Dvir and Amir Shpilka},

title = {Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits},

journal = {SIAM J. COMPUT},

year = {2007},

volume = {36},

number = {5},

pages = {1404--1434}

}

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### Abstract

In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fan-in, that

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Citation Context ... time algorithms then either the Permanent cannot be computed by polynomial size arithmetic circuits or NEXP �⊂ P/poly. The first randomized algorithm for PIT was discovered independently by Schwartz =-=[Sch80]-=- and Zippel [Zip79]. Their well known algorithm simply evaluates the polynomial at a random point and accepts iff the polynomial vanishes at the point. If the polynomial is of degree d and each variab... |

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Citation Context ...t is one of a few problems (and in some sense PIT is the most general problem) for which we have coRP algorithms but no deterministic subexponential time algorithms. Recently Kabanets and Impagliazzo =-=[KI03]-=- suggested an explanation for the lack of algorithms. They showed that efficient deterministic algorithms for PIT imply that NEXP does not have polynomial size arithmetic circuits. Specifically, if PI... |

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Citation Context ... case of codes with two queries (q = 2). Exponential lower bounds were first proved for linear codes [GKST01, Oba02] and then, by techniques from quantum computation, for non-linear codes over GF (2) =-=[KdW03]-=-. The bound of Goldreich et al [GKST01] actually holds for linear LDCs with 2 queries over any finite field, namely that m is at least 2 Ω(n)−log(|F|) , where F is the underlined field. This result is... |

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Citation Context ...depth 3 arithmetic circuits (also known as ΣΠΣ circuits). A ΣΠΣ circuit computes a polynomial of the form k� di � C = Lij(x), (1) i=1 j=1 where the Lij’s are linear functions. Grigoriev and Karpinski =-=[GK98]-=- and Grigoriev and Razborov [GR98] proved exponential lower bounds on the size of ΣΠΣ circuits computing the Permanent and Determinant over finite fields. Over infinite fields exponential lower bounds... |

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Citation Context ...s (in the black box model) [GKS90, BOT88, KS01]. In particular, for the special case of depth 3 circuits with 3 multiplication gates our result resolves an open question asked by Klivans and Spielman =-=[KS01]-=-. 1 Introduction Locally Decodable Codes (LDCs) are error correcting codes that allow the recovery of each symbol of the message from a constant number of entries of the codeword. Polynomial Identity ... |

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Citation Context ...time for this case, thus resolving the question of [KS01]. We note that a complete derandomization is to give a polynomial time algorithm for the problem, as was recently achieved by Kayal and Saxena =-=[KS05]-=-. We discuss their result in the next subsection. 1.5 Recent Results As mentioned above, Kayal and Saxena [KS05] managed to give a polynomial time algorithm for PIT of depth-3 circuits with bounded to... |

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