Maximal arc explained

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

• 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.

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

(q +1 ) - (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(2,2^h), maximal arcs for every degree 2^t, 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 - (q)/(,q - d) (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)}