2024/05/29 by Tushant Mittal, Mittal, Tushant, Sourya Roy +1 · 2 citations
Engineering · #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Fault Detection and Control Systems #Group Theory (math.GR) #Microfluidic and Capillary Electrophoresis Applications #Representation Theory (math.RT) #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.2405.18998
openalex publication_date 2024/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a randomness-efficient homomorphism test in the low soundness regime for functions, f: G→ \mathbbUt, from an arbitrary finite group G to t× 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' U2 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] and Gowers [Comb. Probab. Comput.'08].