If we have to construct Hadamard matrix of order n, and n is power of 2, we can use Kronecker mutiplication of matrices. I have heard, that in case of arbitrary n (divisible by 4, of course) we can somehow use Legendre symbols to construct the matrix. I suppose that we should take the row of Legendre symbols for fixed n (the half of them will be quadratic residues, the won't, so we get half of ones and half of minus ones in the row). But what particular symbols we have to choose? And how to show that the rows will be orthogonal (if they will)?