In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space $H(3,q^2)$, q even, with respect to a symplectic polar space $W(3,q)$ embedded in it.

In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of $\mathrm{PGL}(2,q^2)$, q even, on a non-degenerate conic of $\mathrm{PG}(2,q^2)$ extended in $\mathrm{PG}(2,q^4)$. Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.

在2011年，彭蒂拉（Penttila）和威利福德（Williford）通过探索the极空间线的几何形状，构造了一个无限的新的原始Q多项式3类关联方案，该方案不是由距离正则图产生的$H（3，q^2）$关于辛极空间，甚至q$\mathrm{w\; ^}（3，q）$ 嵌入其中。

在与Penttila和Williford的私人通讯中，田中健三（H. Tanaka）指出，这些方案与Hollmann和Xiang在2006年通过考虑 $\mathrm{聚乳酸}（2，q^2）$，q甚至上的非退化的二次曲线$\mathrm{PG}（2，q^2）$ 扩展到 $\mathrm{PG}（2，q^4）$。因此，出现上述关联方案是否同构的问题。在本文中，我们提供了肯定的答案。作为副产品，我们得到强正则图的同构。