Abstract
An (n,k)affine source over a finite field \( \mathbb{F} \) is a random variable X = (X _{1},..., X _{ n }) ∈ \( \mathbb{F}^n \), which is uniformly distributed over an (unknown) kdimensional affine subspace of \( \mathbb{F}^n \). We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n ^{c} (where c is a large enough constant). Our main results are as follows:

1.
(For arbitrary k): For any n,k and any \( \mathbb{F} \) of size larger than n ^{20}, we give an explicit construction for a function D : \( \mathbb{F}^n \) → \( \mathbb{F}^{k  1} \), such that for any (n,k)affine source X over \( \mathbb{F} \), the distribution of D(X) is ∊close to uniform, where ∊ is polynomially small in \( \mathbb{F} \).

2.
(For k=1): For any n and any \( \mathbb{F} \) of size larger than n ^{c}, we give an explicit construction for a function D: \( \mathbb{F}^n \to \{ 0,1\} ^{(1  \delta )log_2 \mathbb{F}} \), such that for any (n, 1)affine source X over \( \mathbb{F} \), the distribution of D(X) is ∊close to uniform, where ∊ is polynomially small in \( \mathbb{F} \). Here, δ>0 is an arbitrary small constant, and c is a constant depending on δ.
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Research supported by Israel Science Foundation (ISF) grant.
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Gabizon, A., Raz, R. Deterministic extractors for affine sources over large fields. Combinatorica 28, 415–440 (2008). https://doi.org/10.1007/s0049300822593
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Mathematics Subject Classification (2000)
 11T24
 68Q99
 68R05
 05D10