There is an algorithm to construct a polynomial over GF(2) of several variables in algebraic normal form that generates an arbitrary sequence when the variables

There is an algorithm to construct a polynomial over GF(2) of several variables in algebraic normal form that generates an arbitrary sequence when the variables are incremented like a radix-2 binary number. In other words: these are polynomials with n variables. every variable is either zero or one. addition is modulo 2. Each variable corresponds to a single bit in a binary number. Can we construct a polynomial that produces any given length 2^n sequence? One way this is done by induction on the number of variables. I find this proof to be extremely compelling. I derived it myself one time and it has stuck with me. The faster way to do this is through a Walsh transform. This proof converts the problem to a matrix equation by simply manipulating a sum. The Walsh matrix itself is an interesting fractal because it's a Hadamard matrix. These two simple proofs give different perspectives