List Recovery for Random Low-Rate Linear Codes

Abstract

We prove a list recovery guarantee for random low-rate linear codes over sufficiently large prime fields. For fixed dimension $d$, error fraction $\alpha$, and accuracy parameter $\varepsilon$, a random $d$-dimensional linear code $C \subseteq \mathbb{F}_p^n$ is, with high probability, $(\alpha,\ell,\frac{1+\varepsilon}{1-\alpha}\ell)$-list recoverable simultaneously for all input list sizes $\ell\le 2^{O_{\alpha, \varepsilon, d}(n/\log n)}$. The proof is inspired by work of Matou\v{s}ek, P\v{r}\'{\i}v\v{e}tiv\'{y}, and \v{S}kovro\v{n} on reconstructing point sets from their projections. It combines a deterministic graph-theoretic certificate, a nonvanishing determinant criterion, and the Schwartz--Zippel lemma. We also give a lower bound showing that any linear code $C \subseteq \mathbb{F}_p^n$ of dimension at least two cannot be $(\alpha,\ell,\frac{1+\varepsilon}{1-\alpha}\ell)$-list recoverable for feasible list sizes $\ell \geq 2^{\Omega_{\alpha, \varepsilon}(n)}$. In this sense, our result is nearly optimal.