In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code.
More formally, given a code $C \subseteq \Sigma^n$ and input lists $S_1, \hdots, S_n \subseteq \Sigma$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-\rho)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography. Despite its widespread influence, basic, fundamental questions in the study of list recoverability remain open including (1) determining the optimal tradeoff between the error radius $\rho$, the size of the recovered list $L$, and the rate of the code $C$
and (2) constructing explicit codes achieving (or approaching) such tradeoffs.

As our first main result, we establish a tight ``generalized singleton bound'', which exactly characterizes the optimal information-theoretic tradeoff between the error radius $\rho$ and the rate $R$ of the code $C$ in terms of its list-recoverability. Formally, we show that for constant $\ell, L,\rho$ and sufficiently large alphabets $\Sigma$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}\rho$, it is possible for a $(\rho,\ell,L)$ list-recoverable code to have rate $R^*-\epsilon$ but impossible to have rate $R^*+\epsilon$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo (IEEE TIT 2024).

For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL) (FOCS 1995). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes. As a result, a fundamentally new construction technique is needed to explicitly achieve list recovery capacity.