Maximal arc


A maximal arc in a finite projective plane is a largest possible -arc in that projective plane. If the finite projective plane has order q, then for a maximal arc, k, the number of points of the arc, is the maximum possible with the property that no d+1 points of the arc lie on the same line.

Definition

Let be a finite projective plane of order q. Maximal arcs of degree ''d are -arcs in, where k'' is maximal with respect to the parameter d, in other words, k = qd + d − q.
Equivalently, one can define maximal arcs of degree d in as non-empty sets of points K such that every line intersects the set either in 0 or d points.
Some authors permit the degree of a maximal arc to be 1, q or even q + 1. Letting K be a maximal -arc in a projective plane of order q, if d = 1, K is a point of the plane,d = q, K is the complement of a line, andd = q + 1, K is the entire projective plane.
All of these cases are considered to be trivial examples of maximal arcs, existing in any type of projective plane for any value of q. When 2 ≤ d ≤ q − 1, the maximal arc is called non-trivial, and the definition given above and the properties listed below all refer to non-trivial maximal arcs.

Properties

  • The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals. Thus, d divides q.
  • In the special case of d = 2, maximal arcs are known as hyperovals which can only exist if q is even.
  • An arc K having one fewer point than a maximal arc can always be uniquely extended to a maximal arc by adding to K the point at which all the lines meeting K in d − 1 points meet.
  • In PG with q odd, no non-trivial maximal arcs exist.
  • In PG, maximal arcs for every degree 2t, 1 ≤ t ≤ h exist.

Partial geometries

One can construct partial geometries, derived from maximal arcs:
  • Let K be a maximal arc with degree d. 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 :.