Perp-systems and partial geometries

Abstract

Abstract. A perp-system RðrÞ is a maximal set of r-dimensional subspaces of PGðN; qÞ equipped with a polarity r, such that the tangent space of an element of RðrÞ does not intersect any element of RðrÞ. We prove that a perp-system yields partial geometries, strongly regular graphs, two-weight codes, maximal arcs and k-ovoids. We also give some examples, one of them yielding a new pgð8; 20; 2Þ.

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

“…Let S and G be as in the previous subsection, that is, S is a pg (5,5,2) and G is an abelian automorphism group of S acting regularly on the points of S. We will show that G is elementary abelian. Although we know that s = t = 5, we will in this subsection always write s, as we believe that this makes general arguments easier to read.…”

“…This PG-regulus is due to Mathon [5] and is the only known PG-regulus yielding a pg(s, 2(s + 2), 2). Further notice that such a partial geometry has the same parameters as the partial geometry T * 2 (K), which would arise from a maximal 3-arc K in PG(2, q); it is however well known that such a maximal arc does not exist, see Cossu [4], Thas [15], Ball, Blokhuis and Mazzocca [1].…”

“…Although β is an automorphism of G it does not induce an automorphism of our geometry S. It has nevertheless an interesting geometric interpretation. Let S be a partial geometry pg (5,5,2) admitting an abelian Singer group. It is, using the above, easily seen that every two distinct collinear points (elements of G) g and h are contained in two cliques of size 6, namely the one defined by the line L := g, h , and the one defined by M β , where M = g 2 , h 2 , which is a set of 6 points of S no three of which are collinear (note that since D 2 = D, g 2 is indeed collinear with h 2 ).…”

“…Define S to be the geometry with point set the set of elements of GF(81), with line set the set of 6-tuples (b, 1 (81), and with as incidence relation symmetrized containment. Then van Lint and Schrijver have shown that S is a partial geometry pg (5,5,2). It is worthwhile to notice that the point graph of this partial geometry is a cyclotomic graph.…”

Strongly Regular (α, β)-Geometries

“…In [6], a set R of 21 lines in PG(5, 3) is given that is a strongly regular (0, 2)-regulus. The geometry arising from the regulus is a partial geometry with s=8, t=20 and :=2.…”