Edit: I wanted the following matrix to be $W$. Robert Israel suggested I call it $W^T$ instead. I defer to his years of experience and the fact that it gives a better answer to the problem. End Edit.
For 6 times the given matrix, I nominate the following candidate for $W$W^T$
$$\begin{bmatrix}
1 & & & & 1 & & & & 1 \\
1 & & & & & 1 & & 1 & \\
& 1 & & 1 & & & & & 1 \\
& 1 & & & & 1 & 1 & & \\
& & 1 & 1 & & & & 1 & \\
& & 1 & & 1 & & 1 & &
\end{bmatrix}
$$
Does this help?
Gerhard "Ask Me About Binary Matrices" Paseman, 2011.07.28
