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 behave essentially like unstructured random codes for list-decoding from errors above capacity. More precisely, a random linear code $C\subseteq \mathbb{F}_q^n$ of rate $1 - \frac{1}{n}\log_q|B_\rho| + \varepsilon$, where $|B_\rho|$ is the volume of a radius-$\rho$ Hamming ball in $\mathbb{F}_q^n$, satisfies $|C \cap B| = (1\pm o(1)) \frac{|C|\cdot |B|}{q^n}$ simultaneously for all radius-$\rho$ Hamming balls $B$ in $\mathbb{F}_q^n$ with high probability.

This vastly generalizes the previously best known fact that random linear codes of this rate have covering radius at most $\rho n$ with high probability (Blinovsky, 1987).

Second, over prime fields, random linear codes behave essentially like unstructured random codes for zero-error list-recovery, and list-recovery from erasures, above capacity. More precisely, for a prime $q>2$ and input list size $2\leq \ell\leq q-1$, a random linear code $C\subseteq \mathbb{F}_q^n$ of rate $1-\log_q \ell+\varepsilon$ will satisfy $|C \cap S| = (1\pm o(1)) \frac{|C|\cdot \ell^n}{q^n}$ simultaneously for all combinatorial rectangles $S=S_1\times S_2\times\cdots\times S_n$, where $|S_i|=\ell$ for all $i$, with high probability. An analogous result also holds when we can bound $|S_i|$ only for some of the $i$'s.

In particular, we use this to show the abundance of locally leakage-resilient $n$-party linear ramp secret sharing schemes with any linear reconstruction threshold and sublinear threshold gap $O(n/\log n)$ over fields $\mathbb{F}_q$ of polynomial size $q=\Theta(n^\gamma)$ for a constant $\gamma\in(0,1/5)$. Prior work on the existence of leakage-resilient linear secret sharing was stuck at reconstruction thresholds above $n/2$ for both threshold and ramp schemes.

The translate-family result, and hence the list-decoding application, applies over arbitrary finite fields, even when the field size grows with $n$. The list-recovery and leakage applications over prime fields hold under moderate-growth conditions on $q$, for example $q\le n^{1/5-o(1)}$. Our results are obtained through a careful second-moment analysis of the evolution of intersection sizes as random generators are added to $C$ one by one.