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-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)).
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