Constructions of vectorial Boolean functions with maximum component algebraic immunity

Matrices $A$ have been found so that the function $F\colon\mathbb F_2^n\to\mathbb F_2^n$ of the form $F(x)=(f(x),f(Ax),\dots,f(A^{n-1}x))$ where $f$ is the Dalai function in $n=3,4$ variables has the maximal component algebraic immunity. There are no vectorial Boolean functions $F\colon\mathbb F_2^5\to\mathbb F_2^5$ of the form $F(x)=(f(x),f(Ax),f(A^2x)),f(A^3x),f(A^4x))$ 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 $n\times n$. 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 $\operatorname{supp}(f)$ after the action of the linear transformation $A$.