# Legendre symbol for constructing Hadamard matrix

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)?

• It is not known that there is a Hadamard matrix of order $4m$ for all $m$. See en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture – Gerry Myerson May 12 '17 at 9:37
• Yes, I didn't properly state it, but I meant to construct it when it is known that it exists. For example, for n=12. – sooobus May 12 '17 at 9:45
• So perhaps you should edit your question so it says what you meant to say. – Gerry Myerson May 12 '17 at 11:50
• I said, PERHAPS YOU SHOULD EDIT YOUR QUESTION SO IT SAYS WHAT YOU MEANT TO SAY. – Gerry Myerson May 14 '17 at 12:44
• I'm voting to close this question as off-topic because OP has abandoned it. – Gerry Myerson May 16 '17 at 7:10