Self-dual, not self-polar
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The smallest number of points of an incidence structure which is self-dual but not self-polar is 7. For non-binary structures (where a ''point'' may occur more than once in a ''block'') the number is 6.
[1] Peter J. Cameron,et al. Projective and Polar Spaces , 1992 .
[2] Stanislav Jendrol',et al. A non-involutory selfduality , 1989, Discret. Math..