Strongly regular graphs from differences of quadrics (Q1849957)

Suppose in the finite projective space \(\text{PG}(n,q)\) there exists a set \(\mathcal K\) of points such that every hyperplane of \(\text{PG}(n,q)\) contains either \(h\) or \(h'\) points of \(\mathcal K\). It is well known that a strongly regular graph \(\Gamma\) may be obtained from \(\mathcal K\). Embed \(\text{PG}(n,q)\) in \(\text{PG}(n+1,q)\). The vertices of \(\Gamma\) are the points of \(\text{PG}(n+1,q)- \text{PG}(n,q)\), and two vertices are adjacent if and only if the line joining them meets \(\text{PG}(n,q)\) in a point of \(\mathcal K\). Suppose \(p_i\) is some divisor of \(m\). Take a non-degenerate elliptic quadric \(Q^-(2n-1,q^m)\) in \(\text{PG}(2n-1,q^m)\) with quadratic form \(Q\). The underlying vector space \(V(2n,q^m)\) may be considered as a vector space \(V(2nm/p_i,q^{p_i})\) and a non-degenerate elliptic quadric \(Q_i=Q^-(2mn/p_i,q^{p_i})\) can be constructed in \(\text{PG}(2mn/p_i,q^{p_i})\) using the quadratic form \(\text{Tr}_{\text{GF}(q^m)\to \text{GF}(q^{p_i})}\circ Q\). The quadric \(Q_1\) in \(\text{PG}(2mn-1,q)\) with form \(\text{Tr}_{\text{GF}(q^m)\to \text{GF}(q)}\circ Q\) defines a polarity \(\sigma\) in \(\text{PG}(2mn-1,q)\) and the hyperplane \(P^\sigma\) contains either \[ e_i=\frac{(q^{mn-p_i}-1)(q^{mn-1}+1)}{q-1} \text{ or} e_i'=e_i-q^{mn-1} \] points of \(Q_i\) as \(P\in Q_i\) or \(P\notin Q_i\). Note that as a point set in \(\text{PG}(2mn-1,q)\) the quadric \(Q_i\) has size \[ k_i=\frac{(q^{mn-p_i}-1)(q^{mn}+1)}{q-1}. \] Theorem 1. Let \(m,n\geq 2\) be positive itegers and suppose there exist \(l\) distinct integers \(1=p_1,p_2,\dots,p_l\) such that \(p_i\) divides \(p_{i+1}\) for \(i=1,\dots,l-1\) and \(p_l\) divides \(m\). Then there exists a set of size \(k\) with two intersection numbers \(h\) and \(h'=h-q^{mn-1}\) in \(\text{PG}(2mn-1,q)\) with \(h\) as follows: (i) if \(l\) is even, then \(h=e_l-e_{l-1}+e_{l-2}+\cdots+e_2-e_1\) and \(k=k_l-k_{l-1}+k_{l-2}+\cdots+k_2-k_1\); (ii) if \(l\) is odd, then \(h=e_l-e_{l-1}+e_{l-2}+\cdots+e_3-e_2+e_1\) and \(k=k_l-k_{l-1}+k_{l-2}+\cdots+k_3-k_2+k_1\). If instead of using non-degenerate elliptic quadrics we use differences of non-degeterate hyperbolic quadrics essentially the same result may be proved with \[ h_i=\frac{(q^{mn-p_i}+1)(q^{mn-1}-1)}{q-1}, \quad h_i'=h_i+q^{mn-1} \quad \text{and}\quad k_i'=\frac{(q^{mn-p_i}+1)(q^{mn}-1)}{q-1}. \] Theorem 2. Let \(m\geq 2\) and \(n\geq 1\) be positive itegers and suppose there exist \(l\) distinct integers \(1=p_1,p_2,\dots,p_l\) such that \(p_i\) divides \(p_{i+1}\) for \(i=1,\dots,l-1\) and \(p_l\) divides \(m\). Then there exists a set of size \(k\) with two intersection numbers \(h\) and \(h'=h+q^{mn-1}\) in \(\text{PG}(2mn-1,q)\) with \(h\) as follows: (i) if \(l\) is even, then \(h=h_l-h_{l-1}+h_{l-2}+\cdots+h_2-h_1\) and \(k=k_l'-k_{l-1}'+k_{l-2}'+\cdots+k_2'-k_1'\); (ii) if \(l\) is odd, then \(h=h_l-h_{l-1}+h_{l-2}+\cdots+h_3-h_2+h_1\) and \(k=k_l'-k_{l-1}'+k_{l-2}'+\cdots+k_3'-k_2'+k_1'\).