An algorithm for orthogonal designs

Let A =(si) be an n-tuple of positive integers such that Esi = 2k. We give an algorithm which shows that there exists a p = (RA(n, k) - (k+1)) such that there is an orthogonal design of type (2ps1, 2ps2,..., 2psn) in order 2k+p. We evaluate the maximum of p over n-tuples A which add to 2k. Hence we deduce that for any n and k there is an integer q = max RA(n, k) - (k+1) such that for any n-tuple A there is an orthogonal design of type 2qA in order 2q+k.