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Near-Optimal List-Recovery of Linear Code Families

Authors Ray Li , Nikhil Shagrithaya

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ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Theory of computation → Pseudorandomness and derandomization
  • Mathematics of computing → Combinatorics
Keywords
  • Error-Correcting Codes
  • Randomness
  • List-Recovery
  • Reed-Solomon Codes
  • Random Linear Codes

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Abstract

We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least 𝓁^Ω(1/ε), random linear codes of rate R are (1-R-ε, 𝓁, (𝓁/ε)^O(𝓁/ε))-list-recoverable for all R ∈ (0,1) and 𝓁. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all (1-R-ε, 𝓁, L)-list-recoverable linear codes must have L ≥ 𝓁^{Ω(R/ε)}.
Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.

Cite As Get BibTex

Ray Li and Nikhil Shagrithaya. Near-Optimal List-Recovery of Linear Code Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.53

Author Details

Ray Li
  • Department of Mathematics and Computer Science, Santa Clara University, CA, USA
Nikhil Shagrithaya
  • Department of EECS, University of Michigan, Ann Arbor, MI, USA

Funding

  • Li, Ray: Research supported by NSF grant CCF-2347371.
  • Shagrithaya, Nikhil: Research supported in part by NSF grants CCF-2236931 and CCF-2107345.

Acknowledgements

The authors would like to thank Yeyuan Chen and Zihan Zhang for pointing out a mistake in Theorem 12 in an earlier version of the paper.

References

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