Maximal arc
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Maximal arcs are (k,d)-arcs in a projective plane, where k is maximal with respect to the parameter d.
[edit] Definition
Let π be a projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d (
)are (k,d)-arcs in π, where k is maximal with respect to the parameter d or thus k = qd + d − q.
Equivalently, one can define maximal arcs of degree d in π as a set of points K(
) such that every line intersect it either in 0 or d points.
[edit] Properties
- d=q+1 occurs if and only if every point is in K.
- The number of lines through a fixed point p, not on K (provided that
, intersecting K in one point, equals 
. Thus if 
, d divides q - d = 1 occurs if and only if K contains exactly one point.
- d = q occurs if and only if K contains all points except the points on a fixed line.
- In PG(2,q) with q odd, no maximal arcs of degree d with 1 < d < q exist.
- In PG(2,2h), maximal arcs for every degree
exist.
[edit] Partial geometries
One can construct partial geometries, derived from maximal arcs
- Let K be a maximal arc with degree
. 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 : 
. - Consider the space
and let K a maximal arc of degree 
in a two-dimensional subspace π. Consider an incidence structure 
where P contains all the points not in π, B contains all lines not in π and intersecting π in a point in K, and I is again the natural inclusion. 
is again a partial geometry : pg(2h − 1,(2h + 1)(2m − 1),2m − 1).
