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Volume 9 (2013) Article 5 pp. 253-272
Constructing Small-Bias Sets from Algebraic-Geometric Codes
Received: February 21, 2011
Revised: October 29, 2012
Published: February 20, 2013
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Keywords: small-bias sets, algebraic geometry, AG codes, Goppa codes
ACM Classification: F.2.2, G.2
AMS Classification: 94B27, 12Y05

Abstract: [Plain Text Version]

$ \newcommand{\half}{\frac{1}{2}} \newcommand{\eps}{\epsilon} \newcommand{\logeps}{\log({1 \over \epsilon})} \newcommand{\logepsk}{\log({k \over \epsilon})} \newcommand{\set}[1]{{\left\{#1\right\}}} \newcommand{\logepsksquare}{\log^2({k \over \epsilon})} $

We give an explicit construction of an $\eps$-biased set over $k$ bits of size $O\left(\frac{k}{\eps^2 \log(1/\eps)}\right)^{5/4}$. This improves upon previous explicit constructions when $\eps$ 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.

A preliminary version of this paper appeared in FOCS 2009.