We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08].

翻译：本文针对从任意有限群$G$到$t\times t$酉矩阵的函数$f: G\to \mathbb{U}_t$，提出了一种低可靠性机制下的随机高效同态测试方法。我们证明，若此类函数能通过基于小偏差集的去随机化Blum--Luby--Rubinfeld（BLR）测试，则满足以下性质：（i）该函数与由真同态导出的函数存在相关性；（ii）其在低维不可约表示上具有非平凡傅里叶质量。在完全随机机制下，针对有限群上矩阵值函数的此类测试已隐含于Gowers与Hatami [Sbornik: Mathematics '17] 以及Moore与Russell [SIAM Journal on Discrete Mathematics '15] 的研究中。因此，本研究可视为对其结果的近乎最优去随机化。我们的核心技术贡献在于提出“二阶扩展图混合引理”，该引理表明Gowers的$\mathrm{U}^2$范数可通过限制在小偏差子集上进行高效估计。另一推论是推导出Babai、Nikolov与Pyber [SODA'08] 所提出实用引理的“去随机化”版本。