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 [math]\displaystyle{ \pi }[/math] 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 [math]\displaystyle{ \pi }[/math], 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 [math]\displaystyle{ \pi }[/math] 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 [math]\displaystyle{ (q+1)-\frac{q}{d} }[/math]. 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 [math]\displaystyle{ S(K)=(P,B,I) }[/math], 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 : [math]\displaystyle{ pg(q-d,q-\frac{q}{d},q-\frac{q}{d}-d+1) }[/math].
• Consider the space [math]\displaystyle{ PG(3,2^h) (h\geq 1) }[/math] and let K a maximal arc of degree [math]\displaystyle{ d=2^s (1\leq s\leq m) }[/math] in a two-dimensional subspace [math]\displaystyle{ \pi }[/math]. Consider an incidence structure [math]\displaystyle{ T_2^{*}(K)=(P,B,I) }[/math] where P contains all the points not in [math]\displaystyle{ \pi }[/math], B contains all lines not in [math]\displaystyle{ \pi }[/math] and intersecting [math]\displaystyle{ \pi }[/math] in a point in K, and I is again the natural inclusion. [math]\displaystyle{ T_2^{*}(K) }[/math] is again a partial geometry : [math]\displaystyle{ pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\, }[/math].

References

• Ball, S.; Blokhuis, A.; Mazzocca, F. (1997), "Maximal arcs in Desarguesian planes of odd order do not exist", Combinatorica 17: 31–41, doi:10.1007/bf01196129 
• Denniston, R. H. F. (April 1969), "Some maximal arcs in finite projective planes", Journal of Combinatorial Theory 6 (3): 317–319, doi:10.1016/s0021-9800(69)80095-5 
• Hirschfeld, J. W. P. (1979), Projective Geometries over Finite Fields, New York: Oxford University Press, ISBN 978-0-19-853526-3, https://archive.org/details/projectivegeomet0000hirs 
• Mathon, Rudolf (2002), "New maximal arcs in Desarguesian planes", Journal of Combinatorial Theory, Series A 97 (2): 353–368, doi:10.1006/jcta.2001.3218 
• Thas, J. A. (1974), "Construction of maximal arcs and partial geometries", Geometriae Dedicata 3: 61–64, doi:10.1007/bf00181361