Some cardinality estimates for the set of correlation-immune Boolean functions obtained by the mapping AC^w

Abstract

The mapping  was introduced in one of the works devoted to the study of correlation-immune functions. This mapping allows to quickly increase the number of function's variables. It is the base of recursive method of synthesis correlation-immune Boolean functions. However, applying a mapping to the function with all possible parameters allows constructing functions, which is equal to each other. In this paper, there is a criterion of functions equality, which can be produced by  mapping. Also, lower and upper bounds for the set of correlation-immune functions obtained using this mapping are proved. In the proof of these estimates, the function equality criterion is used. The paper presents examples of correlation-immune functions of various weights of a small number of variables and the values of the cardinalities of the corresponding sets, which are obtained by applying the mapping  to the given correlation-immune functions. These results are obtained using computer and they confirm the proved upper and lower bounds for the considered sets of functions.