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Constructing Small-Bias Sets from Algebraic-Geometric Codes

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2 Author(s)
Ben-Aroya, A. ; Blavatnik Sch. of Comput. Sci., Tel-Aviv Univ., Tel-Aviv, Israel ; Ta-Shma, A.

We give an explicit construction of an ¿-biased set over k bits of size O(k/¿2 log(1/¿))5/4This improves upon previous explicit constructions when e is roughly (ignoring logarithmic factors) in the range [k-1.5,k-0.5]. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus. Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of e-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.

Published in:

Foundations of Computer Science, 2009. FOCS '09. 50th Annual IEEE Symposium on

Date of Conference:

25-27 Oct. 2009