Automorphisms Of Quadratic Forms Graphs And Its Applications

Let Qn(F_q) be the set of all quadratic forms in n variables over a finite field F_q of characteristic two. We define the relations on Q_n(F_q) as(x, y) ∈ R_i the type of x — y is iwhere x,y G Q_n(F_q), i = 0,l,2⁺,2^-,3,4⁺,.... These relations form a symmetric association scheme with n + [n/2] associate classes, denoted by X_n. For every relation R_i of x_n we define the graph of quadratic forms Γ_i, which has Q_n(F_q) as the vertex set, two vertices x and y are adjacent whenever the type of x — y is i.In the present paper, we directly determine the automorphisms of the graph Γ₂+ by a matrix skill, and prove thatTheorem A. Suppose n ≥ 2 and q is even. Let Γ₂+ be the graph of quadratic forms in n variables over F_q. Then every automorphism of Γ₂+ is of the formwhere P G GL_n(F_q), a is an automorphism of F_q, and Y ∈ Q(F_q), unless (n,g) = (3,2). When (n,q) = (3,2), the automorphism of Γ₂+ is of the form (1) or is a product of the automorphism of the form (1) and the automorphism of the form (2), whereAs an application of Theorem A, we proved that the automorphisms of Xn is of the form (1).By using the induced subgraphs of the graph of quadratic forms F2+ with n = 2, we construct a classed of new 3—designs, and prove thatTheorem B: There exists a class of simple 3 - (q, \q -1, | (q - 4) (q -6)) designs when q > 8.Similarly, we construct a classed of 3 — (q, \q, $q(q — 4))) designs by using the induced subgraphs of the graph of quadratic forms F2- with n = 2. After deleting the repeated blocks of this class of designs, we obtain a class of simple 3 — (q, \q> \q — 1) designs , this is a class of Hadamard 3—designs.