Maximal arcs are -arcs

Arc (projective geometry)

In mathematics, a -arc in a finite projective plane π is a set of k points of \pi such that each line intersects A in at most d points, and there is at least one line that does intersect A in d points...

 in a projective plane, where k is maximal with respect to the parameter d and the ambient space.

Definition

Let be a projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d ()are -arcs

Arc (projective geometry)

In mathematics, a -arc in a finite projective plane π is a set of k points of \pi such that each line intersects A in at most d points, and there is at least one line that does intersect A in d points...

 in , where k is maximal with respect to the parameter d or thus .

Equivalently, one can define maximal arcs of degree d in as a set of points K() such that every line intersect it either in 0 or d points.

Properties

• occurs if and only if every point is in K.
• The number of lines through a fixed point p, not on K (provided that , intersecting K in one point, equals . Thus if , d divides q
• occurs if and only if K contains exactly one point.
• occurs if and only if K contains all points except the points on a fixed line.
• In with q odd, no maximal arcs of degree d with exist.
• In , maximal arcs for every degree exist.

Partial geometries

One can construct partial geometries

Partial geometry

An incidence structure C= consists of points P, lines L, and flags I \subseteq P \times L where a point p is said to be incident with a line l if \in I...

, derived from maximal arcs

• Let K be a maximal arc with degree . Consider the incidence structure , where P contains all points of the projective plane not on K, B contains all line of the projective plane intersecting K in d points, and the incidence I is the natural inclusion. This is a partial geometry : .
• Consider the space and let K a maximal arc of degree in a two-dimensional subspace . Consider an incidence structure where P contains all the points not in , B contains all lines not in and intersecting in a point in K, and I is again the natural inclusion. is again a partial geometry : .