  Every symplectic spread of PG(3,q), or equivalently every ovoid of Q(4,q), is
shown to give rise to a certain family of permutation polynomials of GF(q) and
conversely. This leads to an algebraic proof of the existence of the
Tits-Luneburg spread of W(2^{2h+1}) and the Ree-Tits spread of W(3^{2h+1}), as
well as to a new family of low-degree permutation polynomials over
GF(3^{2h+1}). We prove the permutation property of the latter polynomials via
an odd characteristic analogue of Dobbertin's approach to uniformly
representable permutation polynomials over GF(2^n). These new permutation
polynomials were later used by Ding, Wang, and Xiang in arXiv:math/0609586 to
produce new skew Hadamard difference sets.
