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-L¨uneburg spread of W(22h+1 ) and the Ree-Tits spread of W(32h+1 ), as well as to a new family of low-degree permutation polynomials over GF(32h+1 ). 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)...

• Type: Conference
• Year: 2003
• Where: FQ
• Authors: Simeon Ball, Michael Zieve
• Keywords: FQ 2003, Non-degenerate Alternating Form, Permutation Polynomials, Symplectic Spread, Theoretical Computer Science