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Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits (2005)
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| Venue: | ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 44 (2005) |
| Citations: | 47 - 14 self |
BibTeX
@MISC{Dvir05locallydecodable,
author = {Zeev Dvir and Amir Shpilka},
title = {Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits},
year = {2005}
}
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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 locally decodable codes 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 2 queries then m = exp(Ω(n)). Previously this was only known for fields of size << 2 n [GKST01]. 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 a locally decodeable code. 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
