Maximal arc

From Wikipedia, the free encyclopedia

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(2,2^h), 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(2^h − 1,(2^h + 1)(2^m − 1),2^m − 1).