Year: 2026
Type: preprint
Abstract: 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 significantly strengthens the best prior results, which follow from work by Blinovsky (1987, 1990) and Levitin (1988) on the covering properties of RLCs. 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2074083719, NNA Track 2: Collaborative Research: Current and future Arctic community vulnerabilities to sea-ice change and economic expansion, G3850789590, G3884855522, G4046065984, Instituto de Telecomunicações, G5475635744, Secure and Efficient Code-Based Cryptography, G6997658380, G7033104259, G766412677
OpenAlex ID: W7165789812