Symplectic Spreads and Permutation Polynomials

Every symplectic spread of PG(3, q), or equivalently every ovoid of Q(4, q), is shown to give a certain family of permutation polynomials of GF (q) and conversely. This leads to an algebraic proof of the existence of the Tits-Lüneburg spread of W (2) and the Ree-Tits spread of W (3), as well as to a new family of low-degree permutation polynomials over GF (3). Let PG(3, q) denote the projective space of three dimensions over GF (q). A spread of PG(3, q) is a partition of the points of the space into lines. A spread is called symplectic if every line of the spread is totally isotropic with respect to a fixed non-degenerate alternating form. Explicitly, the points of PG(3, q) are equivalence classes of nonzero vectors (x0, x1, x2, x3) over GF (q) modulo multiplication by GF (q)∗. Since all non-degenerate alternating forms on PG(3, q) are equivalent (cf. [9, p. 587] or [12, p. 69]), we may use the form ((x0, x1, x2, x3), (y0, y1, y2, y3)) = x0y3 − x3y0 − x1y2 + y1x2. (1) Then a symplectic spread is a partition of the points of PG(3, q) into lines such that (P,Q) = 0 for any points P,Q lying on the same line of the spread. Symplectic spreads are equivalent to other objects. A symplectic spread is a spread of the generalised quadrangle W (q) (sometimes denoted as Sp(4, q)), whose points are the points of PG(3, q) and whose lines are the totally isotropic lines with respect to a non-degenerate alternating form. By the Klein correspondence (see for example [4], [12, pp. 189] or [15]), a spread of W (q) gives an ovoid of the generalised quadrangle Q(4, q) (sometimes denoted O(5, q)) and vice-versa. Let S be a spread of PG(3, q). There are q3 + q2 + q+ 1 points in PG(3, q), and each line contains q + 1 points. Since S is a partition of the points of PG(3, q) into lines, it contains exactly q2 + 1 ∗This author acknowledges the support of the Ministerio de Ciencia y Tecnologia, España.