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By converting the MAXSAT problem to Walsh polynomials, we can efficiently and exactly compute the hyperplane averages of fixed order k. We use this fact to construct initial solutions based on variable configurations that maximize the sampling of hyperplanes with good average evaluations. The Walsh coefficients can also be used to implement a constant time neighborhood update which is integral to a fast next descent local search for MAXSAT (and for all bounded pseudo-Boolean optimization problems.) We evaluate the effect of initializing local search with hyperplane averages on both the first local optima found by the search and the final solutions found after a fixed number of bit flips. Hyperplane initialization not only provides better evaluations, but also finds local optima closer to the globally optimal solution in fewer bit flips than search initialized with random solutions. A next descent search initialized with hyperplane averages is able to outperform several state-of-the art stochastic local search algorithms on both random and industrial instances of MAXSAT.