Vinayak M. Kumar, Ray Li, Jonathan Mosheiff, Gilad Shavit
Thresholds for local properties tell us which guarantees to expect from random linear codes (RLCs) and help us carry those guarantees over to structured codes. A monotone property is $b$-local if a collection of at most $b$ codewords can witness its satisfaction or failure, as in list decoding and list recovery. The threshold of $\mathcal P$ is the code dimension at which the probability that an RLC in $\mathbb{F}_q^n$ satisfies $\mathcal P$ crosses $1/2$ in a sharp transition. We show that the threshold of a $b$-local property $\mathcal P$ is within $O(b)$ dimensions of the expectation threshold. This is the dimension predicted by the best union bound over covers of $\mathcal P$: families of subspaces such that the span of every witness contains at least one member. This gives a general derandomization statement: Any decreasing local property satisfied by RLCs at a given rate holds at a similar rate for pseudorandom ensembles that fool the cover union bound. In addition to previously studied ensembles, these include codes whose parity-check matrix is sampled from a small-bias distribution. Because our threshold characterization applies to all local properties, this derandomization also reaches properties beyond the scope of earlier reductions, including maximum loads in linear hashing for arbitrary input sets, rank-metric list decoding, and list decoding from insertion-deletion errors and burst errors. Our theorem unifies and strengthens prior works on row-symmetric and coordinate-wise linear characterizations, which located the threshold to within $O(q^{2b}\log_q n)$ and $O(b^2+nb/\log_2 q)$ dimensions, respectively. We obtain $O(b)$ error for every $b$-local property, independently of alphabet size. Under suitable pseudorandomness bounds, the rate loss vanishes for every $b=o(n)$, compared to $b=O(\log n)$ for earlier frameworks.
Dean Doron, Tal Leonov, Jonathan Mosheiff, Henrique Navas, Nicolas Resch, João Ribeiro
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.