How did you get this function? Also, @JoeSilverman you are right, the initial value is S(0)=0. Also, I don't have any proof that this should have a solution, so if you can say it doesn't it also works for me :)

Here you can find some info about Hadamard code- en.wikipedia.org/wiki/Hadamard_code Notice that the distance is always half the block length, as required. As explained under the "Construction" category, the generator matrix for the Hadamard code of rank is constructed by listing all the binary strings of length k in lexicographical order as column vectors. The example for k=3 is given in the wiki, and for k=4 you get G = | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | 0