List Recoverable Codes: The Good, the Bad, and the Unknown (hopefully not Ugly)

TL;DR

This survey analyzes list recovery, a broad generalization of decoding from soft information, and maps the landscape of existence, limits, and constructions. It articulates a capacity theorem for random codes, contrasts it with structured code behavior (notably random linear and folded Reed-Solomon codes), and explores distance amplification techniques via AEL constructions. The work also connects list recovery to practical domains such as leakage-resilient secret sharing and pseudorandomness, highlighting both algorithmic progress (e.g., SoS-based decoding) and fundamental barriers (e.g., exponential list sizes at capacity for certain regimes). Overall, the paper synthesizes core tradeoffs between rate, radius, and list size, while charting rich open questions and cross-disciplinary applications.

Abstract

List recovery is a fundamental task for error-correcting codes, vastly generalizing unique decoding from worst-case errors and list decoding. Briefly, one is given ''soft information'' in the form of input lists S_1,...,S_n of bounded size, and one argues that there are not too many codewords that agree a lot with this soft information. This general problem appears in many guises, both within coding theory and in theoretical computer science more broadly. In this article we survey recent results on list recovery codes, introducing both the ''good'' (i.e., possibility results, showing that codes with certain list recoverability exist), the ''bad'' (impossibility results), and the ''unknown''. We additionally demonstrate that, while list recoverable codes were initially introduced as a component in list decoding concatenated codes, they have since found myriad applications to and connections with other topics in theoretical computer science.

Figures (5)

• Figure 1: Left: (Reproduced from resch2020) An illustration of a "puffed-up rectangle" $B(S,\rho)$, created by placing a ball of radius $\rho$ around each point in a combinatorial rectangle $S$. Right: The same rectangle shown with a code $C$ (red vertices). The balls are filtered to show only those with a non-trivial intersection with $C$.
• Figure 2: The hierarchy between the different models. For any edge, the upper model is more general than the lower model, and the edge label denotes the additional relaxation allowed.
• Figure 3: A typical pseudorandom construction strategy (left), and the AEL code construction (right).
• Figure 4: Plots of $1-h_{q,\ell}(x)$ for various values of alphabet $q$ and input list size $\ell$. In blue, $q=19$ and $\ell=1$; in orange, $q=19$ and $\ell=5$; in green, $q=19$ and $\ell=10$; and in red, $q=2048$ and $\ell=8$. Observe that $1-h_{q,\ell}(0) = 1-\log_q\ell$ and $1-h_{q,\ell}(1-\ell/q)=0$, and that $1-h_{q,\ell}$ decreases monotonically between these endpoints. Observe further that when $q$ is very large compared to $\ell$ (cf. the red line), one obtains essentially a straight line with $y$-intercept $1-\log_q\ell$ and $x$-intercept $1-\ell/q$.
• Figure 5: Correspondences between list recovery and expanders, condensers and extractors. Left: The correspondence in \ref{['thm:expander-iff-LR']}. Middle: Another correspondence in guruswami-umans-vadhan-2009-unbalanced-expanders. Bottom: Yet another correspondence in TUZ01.

Theorems & Definitions (13)

• Proposition 2.1: Folklore, cf. guruswami-rudra-2009-multilevel-concatenation
• proof
• Theorem 2.2: alon-edmonds-luby-1995-AEL
• Theorem 3.1: List recovery capacity theorem, cf. resch2020
• proof : Proof of \ref{['thm:LR-capacity-theorem']}
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Theorem 4.7: LS25, adapted to zero-error list recovery