2 research outputs found
Affine extractors over large fields with exponential error
We describe a construction of explicit affine extractors over large finite
fields with exponentially small error and linear output length. Our
construction relies on a deep theorem of Deligne giving tight estimates for
exponential sums over smooth varieties in high dimensions.Comment: To appear in Comput. Comple
Deterministic Extractors for Additive Sources
We propose a new model of a weakly random source that admits randomness
extraction. Our model of additive sources includes such natural sources as
uniform distributions on arithmetic progressions (APs), generalized arithmetic
progressions (GAPs), and Bohr sets, each of which generalizes affine sources.
We give an explicit extractor for additive sources with linear min-entropy over
both and , for large prime , although our
results over require that the source further satisfy a
list-decodability condition. As a corollary, we obtain explicit extractors for
APs, GAPs, and Bohr sources with linear min-entropy, although again our results
over require the list-decodability condition. We further
explore special cases of additive sources. We improve previous constructions of
line sources (affine sources of dimension 1), requiring a field of size linear
in , rather than by Gabizon and Raz. This beats the
non-explicit bound of obtained by the probabilistic method.
We then generalize this result to APs and GAPs