Constant-degree polynomial samplers over F_2^m produce distributions at total variation distance 1-o(1) from Ber(1/3)^⊗n, with concrete bounds for d=1,2,3 and a supporting lemma that no degree-d polynomial has bias exactly 1/3.
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms , pages=
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On Sampling Lower Bounds for Polynomials
Constant-degree polynomial samplers over F_2^m produce distributions at total variation distance 1-o(1) from Ber(1/3)^⊗n, with concrete bounds for d=1,2,3 and a supporting lemma that no degree-d polynomial has bias exactly 1/3.