We show that random linear codes possess nearly optimal discrepancy-type properties in a broad range of settings. Our main results are two general discrepancy theorems: one controls all translates of a fixed test, and the other controls large families of Fourier-pseudorandom tests. Two motivating applications follow:

First, random linear codes ... more >>>

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, ... more >>>

The Gilbert--Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate $\varepsilon^2$ has relative distance at least $\frac{1}{2} - O(\varepsilon)$ with high probability. However, it is a major challenge to construct explicit codes with similar parameters.

One hope to ... more >>>