Maximal arc

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

Definition
Let $$\pi$$ be a finite projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d ( 2 ≤ d ≤ q- 1) are (k,d)-arcs in $$\pi$$, 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 $$\pi$$ 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 (k, d)-arc in a projective plane of order q, if 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.
 * d = 1, K is a point of the plane,
 * d = q, K is the complement of a line (an affine plane of order q), and
 * d = q + 1, K is the entire projective plane.

Properties

 * The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals $$(q+1)-\frac{q}{d}$$. 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(2,q) with q odd, no non-trivial maximal arcs exist.
 * In PG(2,2h), 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 $$S(K)=(P,B,I)$$, 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 : $$pg(q-d,q-\frac{q}{d},q-\frac{q}{d}-d+1)$$.
 * Consider the space $$PG(3,2^h) (h\geq 1)$$ and let K a maximal arc of degree $$d=2^s (1\leq s\leq m)$$ in a two-dimensional subspace $$\pi$$. Consider an incidence structure $$T_2^{*}(K)=(P,B,I)$$ where P contains all the points not in $$\pi$$, B contains all lines not in $$\pi$$ and intersecting $$\pi$$ in a point in K, and I is again the natural inclusion.  $$T_2^{*}(K)$$ is again a partial geometry : $$pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\,$$.