Isomorphism classes of association schemes induced by Hadamard matrices

Every Hadamard matrix $H$ of order $n>1$ induces a graph with $4n$ vertices, called the Hadamard graph $\mathrm{\Gamma }(H)$ of $H$. Since $\mathrm{\Gamma }(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme $(\mathrm{\Omega },S)$ of order $4n$. In this article we deal with fission schemes of $(\mathrm{\Omega },S)$ under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme.