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
* ''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 $(q+1)-\frac$. 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 : $pg(q-d,q-\frac,q-\frac-d+1)$.
* 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 : $pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\,$.

Notes

References

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{{DEFAULTSORT:Maximal Arc
Projective geometry