Spectral analysis of high order correlation immune functions

We consider F₂ⁿ, the vector space of n-tuples of elements from F₂. An n-variable Boolean function is a map from F₂ⁿ into F₂. The weight of a vector x is the number of ones in x and is denoted by |x|. The weight wt(f) of a function f on F₂ⁿ is the number of vectors x on F₂ⁿ such that f(x)=1. A function f is said to be balance if wt(f)=wt(f⊕1)=2^n-1. A subfunction of the Boolean function f is a function f´ obtained by substituting some constants for some variables in f