Implementation complexity of Boolean functions with a small number of ones

Abstract We consider the class Fn, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class Fn, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis {x&y,x‾} $ \{x\&y,\overline x\} $ (or in the basis {x∨y,x‾}) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.