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Fast List Recovery of Univariate Multiplicity Codes

Authors Rohan Goyal , Prahladh Harsha , Mrinal Kumar , Ashutosh Shankar

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Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Design and analysis of algorithms
Keywords
  • list-recovery
  • multiplicity-codes
  • near-linear-algorithm

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Abstract

Recent work gave near-linear time algorithms for list decoding Folded Reed-Solomon codes and univariate multiplicity codes up to capacity in their natural parameter regimes. Unlike most known list decoding algorithms, these techniques appeared inherently tied to list decoding, and it was unclear whether they could be extended to list recovery in near-linear time.
In this work, we resolve this question by giving Õ(n)-time algorithms for list recovery of Folded Reed-Solomon codes and univariate multiplicity codes up to capacity, where n is the block length. Our algorithms build on the lattice-based framework of the prior work, augmented with a new technical ingredient: the construction of suitably structured lattices over the univariate polynomial ring that capture the list recovery problem for these codes.

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Rohan Goyal, Prahladh Harsha, Mrinal Kumar, and Ashutosh Shankar. Fast List Recovery of Univariate Multiplicity Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.67

Author Details

Rohan Goyal
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Mrinal Kumar
  • Tata Institute of Fundamental Research, Mumbai, India
Ashutosh Shankar
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

  • Goyal, Rohan: Supported by (Yael Tauman Kalai’s) grant from Defense Advanced Research Projects Agency (DARPA) under Contract No. HR0011-25-C-0300. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Defense Advanced Research Projects Agency (DARPA).
  • Harsha, Prahladh: Supported by the Department of Atomic Energy, Government of India, under project number RTI400112 and in part by a Google research grant.
  • Kumar, Mrinal: Supported by the Department of Atomic Energy, Government of India, under project number RTI400112, and in part by research grants from Google, SERB and Premji Invest.
  • Shankar, Ashutosh: Supported by the Department of Atomic Energy, Government of India, under project number RTI400112.

References

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