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TR05-044 | 6th April 2005 00:00
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#### Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits

**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 an exponential lower bound on the length of locally decodable codes with 2 queries, over arbitrary fields. Previously such bounds were known for fields of size $<< 2^n$.

2. We show that from every depth 3 arithmetic circuit with a bounded top fan-in, that computes the zero polynomial, one can construct a locally decodable code with 2 queries.

3. As a corollary of the results above we prove a structural theorem for identically zero depth 3 circuits.

4. Using the structural theorem we obtain new PIT algorithms for depth 3 circuits. In particular for such circuits with bounded top fan-in we get

- A deterministic algorithm that runs in quasi-polynomial time

- A probabilistic algorithm that runs in polynomial time and uses only polylogarithmic number of random bits.

In particular for the case of top fan-in = 3 this resolves an open question asked by Klivans and Spielman.