An Approximation to Proof of the Circulant Hadamard Conjecture

Abstract

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 ${\mathbb{Z}}_2^{4m^2}$ to prove that there is no circulant Hadamard matrix in ${\mathbb{Z}}_2^{4m^2}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^2-m}{2}\le a\le \frac{3m^2-m}{2}$ and $b=2m^2-m-a$ and with $\frac{m^2+m}{2}\le 2m^2-a\le \frac{3m^2+m}{2}$ and $b=m+a$.

7 pages, substantial text overlap eliminated