Let \(\Gamma\) be a distance-regular graph of diameter 3 with a strongly regular graph \(\Gamma_{3}\). Finding the parameters of \(\Gamma_{3}\) from the intersection array of \(\Gamma\) is a direct problem, and finding the intersection array of \(\Gamma\) from the parameters of \(\Gamma_{3}\) is its inverse. The direct and inverse problems were solved by A.A. Makhnev and M.S. Nirova: if a graph \(\Gamma\) with intersection array \(\{k,b_{1},b_{2};1,c_{2},c_{3}\}\) has eigenvalue \(\theta_{2}=-1\), then the graph complementary to \(\Gamma_{3}\) is pseudo-geometric for \(pG_{c_{3}}(k,b_{1}/c_{2})\). Conversely, if \(\Gamma_{3}\) is a pseudo-geometric graph for \(pG_{\alpha}(k,t)\), then \(\Gamma\) has intersection array \(\{k,c_{2}t,k-\alpha+1;1,c_{2},\alpha\}\), where \(k-\alpha+1\leq c_{2}t<k\) and \(1\leq c_{2}\leq\alpha\). Distance-regular graphs \(\Gamma\) of diameter 3 such that the graph \(\Gamma_{3}\) (\(\bar{\Gamma}_{3}\)) is pseudogeometric for a net or a generalized quadrangle were studied earlier. In this paper, we study intersection arrays of distance-regular graphs \(\Gamma\) of diameter 3 such that the graph \(\Gamma_{3}\) (\(\bar{\Gamma}_{3}\)) is pseudogeometric for a dual 2-design \(pG_{t+1}(l,t)\). New infinite families of feasible intersection arrays are found: \(\{m(m^{2}-1),m^{2}(m-1),m^{2};1,1,(m^{2}-1)(m-1)\}\), \(\{m(m+1),(m+2)(m-1),m+2;1,1,m^{2}-1\}\), and \(\{2m(m-1),(2m-1)(m-1),2m-1;1,1,2(m-1)^{2}\}\), where \(m\equiv\pm 1\) (mod 3). The known families of Steiner 2-designs are unitals, designs corresponding to projective planes of even order containing a hyperoval, designs of points and lines of projective spaces \(PG(n,q)\), and designs of points and lines of affine spaces \(AG(n,q)\). We find feasible intersection arrays of a distance-regular graph \(\Gamma\) of diameter 3 such that the graph \(\Gamma_{3}\) (\(\bar{\Gamma}_{3}\)) is pseudogeometric for one of the known Steiner 2-designs.

令 \(\Gamma\)是一个直径为 3 的距离正则图，带有一个强正则图 \(\Gamma_{3}\)。从\(\Gamma\)的交集数组中 求\(\Gamma_{3}\)的参数 是一个直接的问题，从\(\Gamma_{ )的参数求 \(\ Gamma\)的交集数组 3}\)是它的倒数。AA Makhnev 和 MS Nirova 解决了正反问题：如果一个图 \(\Gamma\)与交集数组\(\{k,b_{1},b_{2};1,c_{2},c_ {3}\}\)具有特征值\(\theta_{2}=-1\)，那么与\(\Gamma_{3}\)互补的图 是伪几何的\(pG_{c_{3}}(k,b_{1}/c_{2})\)。相反，如果 \(\Gamma_{3}\)是\(pG_{\alpha}(k,t)\)的伪几何图，则 \(\Gamma\)具有交集数组\(\{k, c_{2}t,k-\alpha+1;1,c_{2},\alpha\}\)，其中\(k-\alpha+1\leq c_{2}t<k\)和\( 1\leq c_{2}\leq\alpha\)。直径为 3 的距离正则图 \(\Gamma\)使得图\(\Gamma_{3}\) ( \(\bar{\Gamma}_{3}\) ) 对于网络或广义四边形之前研究过。在本文中，我们研究了直径为 3的距离正则图\(\Gamma\)的交集数组 ，使得图\(\Gamma_{3}\) ( \(\bar{\Gamma}_{3}\) ) 是双 2-design \(pG_{t+1}(l,t)\) 的伪几何。找到了新的可行交集数组的无限族：\(\{m(m^{2}-1),m^{2}(m-1),m^{2};1,1,(m^{ 2}-1)(m-1)\}\) , \(\{m(m+1),(m+2)(m-1),m+2;1,1,m^{2} -1\}\)和\(\{2m(m-1),(2m-1)(m-1),2m-1;1,1,2(m-1)^{2}\} \)，其中\(m\equiv\pm 1\) (mod 3)。Steiner 2-designs 的已知族是 unitals ，对应于包含超椭圆的偶数阶射影平面的设计，射影空间\(PG(n,q)\)的点和线的设计，以及仿射点和线的设计空格 \(AG(n,q)\). 我们找到直径为 3的距离正则图\(\Gamma\)的可行交集数组 ，使得图\(\Gamma_{3}\) ( \(\bar{\Gamma}_{3}\) ) 是已知 Steiner 2 设计之一的伪几何。