(Both the published version and the arXiv version are available online.)

Let  X  denote the three-dimensional projective space over  F_q.  A spread of  X  is a partition of the points of  X  into lines. Such a spread is symplectic if there is a non-degenerate alternating form  <∗,∗>  on  X  such that  <u,v> = 0  for any points  u,v  which lie on the same line of the spread. We show that every symplectic spread gives rise to a certain family of permutation polynomials of  F_q,  and conversely one can construct a symplectic spread from such a family of permutation polynomials. We apply this to the six known families of symplectic spreads: five of these families correspond to well-known permutation polynomials, which yields simple new verifications that these are indeed symplectic spreads.

The sixth known family of symplectic spreads was discovered by Kantor as an ovoid of the generalized quadrangle  Q(4,q),  by taking a slice of the Ree-Tits ovoid of  Q(6,q).  By our result, the fact that this yields a symplectic spread is equivalent to the assertion that  x^2r+3+(cx)^r-c²x  permutes the field  F  of cardinality  r²/3  whenever  r  is a power of  3  and  c  lies in  F.  These permutation polynomials have the unusual property that their degree is on the order of  (#F)^½,  while the polynomials are not exceptional when  r > 3  and  c≠0;  these are the only known non-exceptional permutation polynomials of  F_q  having degree on the order of  q^½  in case  q  is an odd nonsquare. The fact that these polynomials permute  F  follows from Kantor's result; conversely, we use a variant of Dobbertin's method to show directly that these polynomials permute F, which in turn yields a new proof of Kantor's result that is much simpler than the original proof.

Additional comment from May 2007:  Our new permutation polynomials have been used by Ding, Wang and Xiang to construct a new family of skew Hadamard difference sets.

Michael Zieve:  home page   publication list