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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 $\mathbb{Z}_p$, (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 $\mathrm{GL}_n(\mathbb{F}_q)$, and finite-dimensional Lie algebras over $\mathbb{F}_q$. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between $\mathbb{F}_q^n$ and $\mathbb{F}_q$.
Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as $\mathbb{F}_q^n$). 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 $O(\varepsilon^{-2})$ (for agreement parameter $\varepsilon$). This improves upon the currently best-known bound of $O(\varepsilon^{-105})$ due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

Submission history

From: Tushant Mittal [view email]
[v1] Sun, 7 Sep 2025 00:00:35 UTC (63 KB)
[v2] Mon, 27 Jul 2026 02:09:02 UTC (61 KB)