Best Affine Approximations of Boolean Functions and Applications to Low Order Approximations

Low order approximations of Boolean functions are studied in this paper. In particular, best affine approximations of quadratic functions are analyzed using Dickson theorem, leading to an explicit formula for their direct computation, without using the Walsh transform. Expressions to determine all the best affine approximations of linear combinations of quadratic functions are proved. The tools developed are suitable to determining low order approximations; they are applied to certain low degree functions with arbitrary number of variables and allow to efficiently derive all of their best quadratic approximations.