  Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field
$\mathbb{F}$. The polynomial is said to be unbiased if the distribution of
$f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform
distribution over $\mathbb{F}$, and is called biased otherwise. The polynomial
is said to have low rank if it can be expressed as a composition of a few lower
degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and
Lovett [FOCS 2008] showed that bias implies low rank for fixed degree
polynomials over fixed prime fields. This lies at the heart of many tools in
higher order Fourier analysis. In this work, we extend this result to all prime
fields (of size possibly growing with $n$). We also provide a generalization to
nonprime fields in the large characteristic case. However, we state all our
applications in the prime field setting for the sake of simplicity of
presentation.
  Using the above generalization to large fields as a starting point, we are
also able to settle the list decoding radius of fixed degree Reed-Muller codes
over growing fields. The case of fixed size fields was solved by Bhowmick and
Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman
[STOC 2008]. Here, we show that the list decoding radius is equal the minimum
distance of the code for all fixed degrees, even when the field size is
possibly growing with $n$.
  Additionally, we effectively resolve the weight distribution problem for
Reed-Muller codes of fixed degree over all fields, first raised in 1977 in the
classic textbook by MacWilliams and Sloane [Research Problem 15.1 in Theory of
Error Correcting Codes].
