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 Parity requires $2^{n^{\Omega(1)}}$ size De Morgan circuits of constant depth using a new method which is completely “top-down” in the sense of [HJP95]. The proof relies crucially on the core ideas developed in a line of work [HJP95, PPZ99, MW19, GRSS24] which previously established top-down lower bounds ... more >>>

Robust Sunflower lemmas imply that any large enough monotone DNF of width $w$ contains a sunflower, i.e., a DNF equivalent to the conjunction of a common core with a DNF that is heavily biased towards $1$. While these lemmas are typically proved in the context of the uniform distribution or ... more >>>