  Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$
must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use
the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there
is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for
sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq
a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq
2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$.

Combinatorics;Group Theory