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 and the ambient space.

[edit] Definition

Let π be a projective plane of order q (not necessarily desarguesian). Maximal arcs of degree d (d\geq 1)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(\neq \emptyset) 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 d\neq (q+1)) , intersecting K in one point, equals (q+1)-\frac{q}{d}. Thus if d\leq q , 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 2^t,1\leq t\leq h exist.

[edit] Partial geometries

One can construct partial geometries, derived from maximal arcs

  • Let K be a maximal arc with degree 1<d\leq q. 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}{d},q-\frac{q}{d}-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 π. Consider an incidence structure T_2^{*}(K)=(P,B,I) 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. T_2^{*}(K) is again a partial geometry : pg(2h − 1,(2h + 1)(2m − 1),2m − 1).