We leverage proof techniques Fourier analysis and an existing result in coding theory to derive new bounds for the problem of non-interactive simulation of binary random variables. Previous bounds in the literature were derived by applying data processing inequalities concerning maximal correlation or hypercontractivity. We show that our bounds are sharp in some regimes. For a specific instance of problem parameters, our main result answers an open problem posed by E. Mossel in 2017. As by-products of our analyses, various new properties of the average distance and distance enumerator of binary block codes are established.
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