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 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 , 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.[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 . 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.[2]
- In PG(2,q) with q odd, no non-trivial maximal arcs exist.[3]
- In PG(2,2h), maximal arcs for every degree 2t, 1 ≤ t ≤ h exist.[4]
Partial geometries
One can construct partial geometries, derived from maximal arcs:[5]
- 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 : .
Notes
- ↑ Hirschfeld 1979, pp. 325
- ↑ Hirschfeld 1979, pg. 328
- ↑ Ball, Blokhuis & Mazzocca 1997
- ↑ Denniston 1969
- ↑ Thas 1974
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, MR 98h:51014, Zbl 0880.51003
- Denniston, R.H.F. (1969), "Some maximal arcs in finite projective planes", J. Combin. Theory 6: 317–319, doi:10.1016/s0021-9800(69)80095-5, MR 39#1345, Zbl 0167.49106
- Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, New York: Oxford University Press, ISBN 0-19-853526-0
- Mathon, R. (2002), "New maximal arcs in Desarguesian planes", J. Combin. Theory Ser. A 97: 353–368, doi:10.1006/jcta.2001.3218, MR 2002k:51012, Zbl 01763788
- Thas, J.A. (1974), "Construction of maximal arcs and partial geometries", Geom. Dedicata 3: 61–64, doi:10.1007/bf00181361, MR 50#1931, Zbl 0285.50018