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

Notes

1. Hirschfeld 1979, pp. 325
2. Hirschfeld 1979, pg. 328
3. Ball, Blokhuis & Mazzocca 1997
4. Denniston 1969
5. 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 1466573, Zbl 0880.51003
• Denniston, R.H.F. (1969), "Some maximal arcs in finite projective planes", J. Comb. Theory, 6 (3): 317–319, doi:10.1016/s0021-9800(69)80095-5, MR 0239991, Zbl 0167.49106
• Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, New York: Oxford University Press, ISBN 978-0-19-853526-3
• Mathon, R. (2002), "New maximal arcs in Desarguesian planes", J. Comb. Theory A, 97 (2): 353–368, doi:10.1006/jcta.2001.3218, MR 1883870, Zbl 1010.51009
• Thas, J.A. (1974), "Construction of maximal arcs and partial geometries", Geom. Dedicata, 3: 61–64, doi:10.1007/bf00181361, MR 0349437, Zbl 0285.50018