Abstract.
We prove that the sum of d small-bias generators \(L : {\mathbb{F}}^{s} \rightarrow {\mathbb{F}}^{n}\) fools degree-d polynomials in n variables over a field \({\mathbb{F}}\), for any fixed degree d and field \({\mathbb{F}}\), including \({\mathbb{F}} = {\mathbb{F}}_{2} = \{0, 1\}\). Our result builds on, simplifies, and improves on both the work by Bogdanov and Viola (FOCS ’07) and the follow-up by Lovett (STOC ’08). The first relies on a conjecture that turned out to be true only for some degrees and fields, while the latter considers the sum of 2d small-bias generators (as opposed to d in our result).
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Manuscript received 1 September 2008
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Viola, E. The Sum of D Small-Bias Generators Fools Polynomials of Degree D. comput. complex. 18, 209–217 (2009). https://doi.org/10.1007/s00037-009-0273-5
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DOI: https://doi.org/10.1007/s00037-009-0273-5