Maximal arc

Maximal arcs are (k,d)-arcs in a projective plane, where k is maximal with respect to the parameter d and the ambient space.

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.

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(2,2^h), maximal arcs for every degree $2^t,1\leq t\leq h$ exist.

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(2^h − 1,(2^h + 1)(2^m − 1),2^m − 1).