The complete list of maximal cliques of \(\text{Quad}(n,q)\), \(q\) even (Q1279871)

Let \(q\) be a prime power, \(F\) be a field of order \(q\) and \(V\) be a vector space of finite dimension \(n\) over \(F\). We define the distance-regular graph \(Q= \text{Quad} (n,q)\) as follows: the vertex set of \(Q\) is the set of all quadratic forms on \(V\), with forms \(x\) and \(y\) adjacent in \(Q\) if the rank \(\text{rk} (x-y)=1\) or 2. The problem that concerns in this paper is that of classifying the maximal cliques of \(Q\), where \(q\) is even and \(q\geq 4\). In the cases \(q=2\) or \(q\) odd, the complete classification is known. The five types of maximal cliques are also known: two types of grand cliques (of size \(q^n\)), cubic cliques (of size \(q^3\)), quadratic cliques (of size \(q^2+1\)) and linear cliques (of order \(3q\) or \(3q-2\)). Theorem 10. Let \(C\) be a maximal clique of \(\text{Quad} (n,q)\) with \(q\) even and \(q\geq 4\). Further suppose that \(C\) is not a grand, cubic, quadratic or linear clique. Then \(| C| =8\). Moreover, such cliques exist for all \(n,\;q\).