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Optimal Testing of Reed-Muller Codes with an Online Adversary

Authors Esty Kelman , Uri Meir , Kai Zhe Zheng

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LIPIcs.CCC.2026.31.pdf
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ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Theory of computation → Error-correcting codes
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property testing
  • Low degree testing

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Abstract

Motivated by applications to property testing in the online-erasure model of Kalemaj, Raskhodnikova, and Varma (ITCS 2022 and Theory of Computing 2023), we define and analyze semi-sample-based testers for Reed-Muller codes. The task in Reed-Muller testing is to determine whether an input function f: 𝔽ⁿ → 𝔽 belongs to the Reed-Muller code or is far from it, using as few point queries to f as possible. Reed-Muller testing is a well-studied task with its roots in both the Property Testing and Probabilistically Checkable Proofs literature. The online-erasure model introduces a twist: after each query made, an adversary may erase up to t points of the input function, potentially thwarting any test in which the queries follow a predictable pattern.
Semi-sample-based testers are a hybrid between sample-based testers - which can only make uniformly random queries to the input function - and standard testers, which can choose their queries freely. They are designed with the online-erasure model in mind and operate by first choosing some subset S of the domain and then making their queries uniformly at random inside of S. We describe semi-sample-based testers for the Reed-Muller code and give an optimal analysis of their soundness. 
Consequently, we show that semi-sample-based testers are indeed effective in the presence of online erasures, and thereby achieve optimal query complexity for testing the Reed-Muller code in the online-erasure model. This result improves upon prior work of Minzer and Zheng (SODA 2024). As an added bonus, we show that semi-sample-based testers also exist for the lifted affine-invariant codes of Guo, Kopparty, and Sudan (ITCS 2013), thereby providing the first known testers for these codes in the online-erasure model.

Cite As Get BibTex

Esty Kelman, Uri Meir, and Kai Zhe Zheng. Optimal Testing of Reed-Muller Codes with an Online Adversary. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 31:1-31:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.31

Author Details

Esty Kelman
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
  • Boston University, Cambridge, MA, USA
Uri Meir
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Kai Zhe Zheng
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Kelman, Esty: Supported in part by NSF TRIPODS program (award DMS-2022448), NSF Grant No. 2022446, and NSF CCF award 2227876.
  • Zheng, Kai Zhe: Supported by the NSF GRFP DGE-2141064.

Acknowledgements

Part of this research was carried out during the authors’ visit to the Simons Institute for the Theory of Computing at UCB. The authors wish to thank Pooya Hatami for bringing [Alexander Golovnev et al., 2024] and the references therein to our attention; Corollary 1.4 there essentially proves Theorem 3 by generalizing the lower from [Ben-Eliezer et al., 2024] to all fields, showing that our new algorithm is optimal.

References

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