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.
Let
π
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
π
, 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
π
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.
The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals
(
q
+
1
)
−
q
d
. 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(2,2h), maximal arcs for every degree 2t, 1 ≤ t ≤ h exist.
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 :
p
g
(
q
−
d
,
q
−
q
d
,
q
−
q
d
−
d
+
1
)
.
Consider the space
P
G
(
3
,
2
h
)
(
h
≥
1
)
and let K a maximal arc of degree
d
=
2
s
(
1
≤
s
≤
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 :
p
g
(
2
h
−
1
,
(
2
h
+
1
)
(
2
m
−
1
)
,
2
m
−
1
)
.
