Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes

List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit constructions which can be efficiently list-recovered up to capacity, namely a fraction of errors approaching $1-R$ where $R$ is the code rate. Chen and Zhang and related works showed that folded Reed-Solomon codes and linear codes must have list sizes exponential in $1/\epsilon$ for list-recovering from an error-fraction $1-R-\epsilon$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/\epsilon$. In contrast to such limitations, we show, extending algorithmic advances of Ashvinkumar, Habib, and Srivastava for list decoding, that even if the lists in the case of list-recovery are large, they are highly structured. In particular, we can output a compact description of a set of size only $\ell^{O((\log \ell)/\epsilon)}$ which contains the relevant list, while running in time only polynomial in $1/\epsilon$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/\epsilon}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery.