Does a nice 430-cap of PG(6, 4) exist?

There are three partial quadrangles which arise from particularly nice caps of projective spaces, but there is one curious possibility left over. This missing piece in the puzzle was uncovered by Rob Calderbank almost thirty years ago. First I will explain how we might obtain a partial quadrangle from linear representation.

Suppose we have a geometry S of points and lines which form a partial linear space of order (s,t) (s+1 points on any line and t+1 lines incident with any point). A so-called linear representation of S into an affine space AG(n+1,s+1) is an embedding such that the line set S is mapped to a union of parallel classes of lines of AG(n+1,s+1). These lines of S then define in the hyperplane at infinity a set of points K of size t+1 (by intersection).

Example (Linear representations of ovoids of PG(3,q)):

Consider an ovoid \mathcal{O} of PG(3,q), and embed PG(3,q) into PG(4,q) as a hyperplane \Pi. This gives us an affine space AG(4,q), where the points of AG(4,q) are the points of PG(4,q) not in \Pi, and the lines of AG(4,q) are the lines of PG(4,q) not contained in \Pi. So there are q^4 affine points. Now consider the following geometry:

  • Points: affine points
  • Lines: affine lines meeting \Pi in a point of \mathcal{O}.

This produces a partial linear space of order (q-1,q^2). Moreover, it is a partial quadrangle known as T^*_3(\mathcal{O}) and was first constructed by Peter Cameron.

More generally, if K is a set of points of PG(n,q) and if its linear representation is a partial quadrangle, then it is not difficult to prove that K has to be a (t+1)-cap of PG(n,q) such that there is a constant \mu such that each point not on K is incident with t+1-\mu tangents. Calderbank (1982) almost gave a complete classification of partial quadrangles which arise from linear representation. (Based on number-theoretic techniques!). Moreover, he listed the possible parameters of the associated strongly regular graph:

  • n = 3 and K is an ovoid of PG(3,q) (as above);
  • K is an 11-cap of PG(4,3): s = 2, t = 10, \mu = 2. (There is an example due to Coxeter).
  • K is the unique 56-cap of PG(5,3): s = 2, t = 55, \mu = 20. (First discovered by Segre).
  • K  is a 78-cap of PG(5,4): s=3, t=77, \mu=14. (There is an example by Hill).
  • K is a 430-cap of PG(6,4): s=3, t=429, \mu=110. (No such cap has yet been found).

Tsanakis and Wolfskill (1987) completed Calderbank’s classification by proving that if q>4, then K is an ovoid of PG(3,q).

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