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A General Framework for Low Soundness Homomorphism Testing

Authors Tushant Mittal , Sourya Roy

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LIPIcs.ITCS.2026.103.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Property Testing
  • Coding Theory

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Abstract

We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. 
In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and ℤ_p, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of GL_n(F_q), and finite-dimensional Lie algebras over F_q. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between F_qⁿ and F_q. 
Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as F_qⁿ). No tests were known for non-abelian groups.
As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of O(ε^{-2}) (for agreement parameter ε). This improves upon the currently best-known bound of O(ε^{-105}) due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

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Tushant Mittal and Sourya Roy. A General Framework for Low Soundness Homomorphism Testing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 103:1-103:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.103

Author Details

Tushant Mittal
  • Stanford University, CA, USA
Sourya Roy
  • University of Iowa, Iowa City, IA, USA

Funding

  • Mittal, Tushant: T.M. is a postdoctoral fellow supported by the NSF grants CCF-2143246 and CCF-2133154.
  • Roy, Sourya: S.R. was partially supported by the Old Gold Summer Fellowship from The University of Iowa.

Acknowledgements

We sincerely thank the reviewer for their helpful comments and careful reading of our manuscript.

References

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