Constructions of vectorial boolean functions with maximum component algebraic immunity

Matrices A have been found so that the function F : Fn ^ Fn of the form F(x) = (f (x), f (Ax), ..., f (A^n-1x)) where f is the Dalai function in n = 3, 4 variables has the maximal component algebraic immunity. There are no vectorial Boolean functions F : F^ ^ F2 of the form F(x) = (f (x), f (Ax), f (A²x)), f (A³x), f (A⁴x)) with the maximal component algebraic immunity where f is the Dalai function in 5 variables. Let f be a Boolean function with the maximal algebraic immunity in an odd number n of variables and A be a non-degenerate matrix n x n. Then the function g(x) = f (x) + f (Ax) has the maximal algebraic immunity only if exactly half of the set supp(f) remains in the set supp(f) after the action of the linear transformation A.

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