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Abstract:We prove that random linear codes have nearly optimal discrepancy properties in a broad range of regimes. Our main results are two general theorems: one controlling all translates of a fixed test, and another controlling large families of Fourier-pseudorandom tests. Two motivating applications follow.
First, random linear codes match unstructured random codes for list-decoding from errors above capacity. If $C\subseteq\mathbb F_q^n$ is a random linear code of rate $1-\frac1n\log_q |B_\rho|+\epsilon$, where $B_\rho$ is a radius-$\rho$ Hamming ball, then with high probability $$ |C\cap B|=(1\pm o(1))\frac{|C||B|}{q^n} $$ simultaneously for all radius-$\rho$ Hamming balls $B\subseteq\mathbb F_q^n$. This extends the classical result that such codes have covering radius at most $\rho n$ whp (Blinovsky, 1987).
Second, over prime fields, random linear codes match unstructured random codes for zero-error list-recovery above capacity. For prime $q>2$ and $2\le \ell\le q-1$, a random linear code of rate $1-\log_q\ell+\epsilon$ satisfies, with high probability, $$ |C\cap S|=(1\pm o(1))\frac{|C|\ell^n}{q^n} $$ simultaneously for all rectangles $S=S_1\times\cdots\times S_n$ with $|S_i|=\ell$. As a consequence, there are abundant $n$-party linear ramp secret sharing schemes over $\mathbb F_q$ with privacy threshold about $n/(2\log q)$ and reconstruction threshold about $5n/(2\log q)$, resilient to balanced local leakage; prior existence results required thresholds above $n/2$ even in this case.
The translate result, hence the list-decoding application, holds over arbitrary finite fields, even growing with $n$. The list-recovery and leakage applications hold over prime fields under moderate growth, e.g. $q\le n^{1/5-o(1)}$. The proofs use a refined second-moment analysis tracking intersection sizes as random generators are added to $C$.

Submission history

From: Jonathan Mosheiff [view email]
[v1] Tue, 23 Jun 2026 12:01:54 UTC (53 KB)