We show that random linear codes (RLCs) 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 examples follow: First, RLCs behave essentially like unstructured random codes for list-decoding from errors above capacity. More precisely, an RLC $C\subseteq \mathbb{F}_q^n$ of rate $1 - \frac{1}{n}\log_q|B_ρ| + \varepsilon$, where $|B_ρ|$ is the volume of a radius-$ρ$ 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-$ρ$ Hamming balls $B$ with high probability. This vastly generalizes the previously best known fact that RLCs of this rate have covering radius at most $ρn$ with high probability (Blinovsky, 1987). Second, over prime fields, RLCs 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 $2\leq \ell\leq q-1$, an RLC $C\subseteq \mathbb{F}_q^n$ 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. 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 of polynomial size $q=Θ(n^γ)$ for a constant $γ\in(0,1/5)$. Prior work was stuck at reconstruction thresholds above $n/2$ for both threshold and ramp schemes.

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