On Hyperfocused Arcs in PG(2,q)

0.16 MB·English

by  Massimo Giulietti

ON HYPERFOCUSED ARCS IN PG(2,q)
M. GIULIETTI AND E. MONTANUCCI
Abstract. Ak-arcinaDearguesianprojectiveplanewhosesecantsmeetsomeexternal
6
0 line in k 1 points is said to be hyperfocused. Hyperfocused arcs are investigated in
−
0 connection with a secret sharing scheme based on geometry due to Simmons. In this
2 paper it is shown that point orbits under suitable groups of elations are hyperfocused

n arcs with the significant property of being contained neither in a hyperoval, nor in a
a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose
J
secants admit a blocking set of minimum size, is introduced: a construction method is

0
provided, together with the classification for size up to 10.
2

]
O
C
1. Introduction
.
h
t
a Hyperfocused arcs were introduced in connection with a secret sharing scheme based
m
on geometry due to Simmons [11]. The implementation of this scheme needs an arc
[
in a Desarguesian projective plane with the property that its secant lines intersect some

1 externallineinaminimalnumber ofpoints. Simmonsonlyconsidered planesofoddorder,
v
where this minimal number equals the number of points of the arc [4]. He introduced the
8
8 term sharply focused set for arcs satisfying the aforementioned property. Sharply focused
4
sets in Desarguesian projective planes of odd order were classified by Beutelspacher and
1
0 Wettl [3], whose result was based on a previous paper by Wettl [12].
6
0 In 1997 Holder [9] extended Simmons’s investigation to Desarguesian planes of even order.
/ In such planes the secants of an arc of size k may meet an external line in only k 1
h
−
t points, yet the classification of arcs having this property seems to be an involved problem.
a
m Holder used the term super sharply focused sets for such arcs and gave some constructions
: for them.
v
i In a recent paper [5], Cherowitzo and Holder proposed the term hyperfocused arc instead
X
of super sharply focused set. They provided the classification of small hyperfocused arcs,
r
a and constructed new examples, one of which gave a negative answer to a question raised
by Drake and Keating [7] on the possible sizes of a hyperfocused arc.
Some open problems were pointed out by Cherowitzo and Holder, including the existence
of hyperfocused arcs which are neither contained in a proper subplane nor in a hyperoval.
In this paper a positive answer to this question is given. The main toolis the investigation
2000 Math. Subj. Class.: Primary 51E21,Secondary 51A30, 05B25, 05C70.
Keywords: Desarguesianplane, arc, blocking set, 1-factorization,secret sharing scheme.
ThisresearchwasperformedwithintheactivityofGNSAGAoftheItalianINDAM,withthefinancial
supportoftheItalianMinistryMIUR,project“Strutturegeometriche,combinatoricaeloroapplicazioni”,
PRIN 2004-2005.
1
2 GIULIETTI ANDMONTANUCCI
of the so-called translation arcs, i.e. arcs which are point orbits under a group of elations.
In Section 3 it is shown that such arcs are hyperfocused, and it is proved that sometimes
they are contained neither in a hyperoval nor in a proper subplane, see Theorem 3.8.
The concept of hyperfocused arc can be naturally extended to that of generalized hyper-
focused arc, that is an arc of size k for which there exists an external point set of size k 1
−
meeting eachofitssecants. Recently, Aguglia,Korchm´arosandSiciliano[1]provedthatin
Desarguesian planes of even order any generalized hyperfocused arc is hyperfocused, pro-
vided that it is contained in a conic. In Section 4 we provide a construction of generalized
hyperfocused arcs which are not hyperfocused. Also, a classification of small general-
ized hyperfocused arcs is proved using the graph-theoretic concept of 1-factorizations of
a complete graph, see Section 5.
2. Definitions and Notation
Let PG(2,q) be the Desarguesian plane over F , the finite field with q elements. A k-arc
q
in PG(2,q) is a set of k points no three of which are collinear. Any line containing two
K
points of is said to be a secant of . A blocking set of the secants of is a point set
K K K
PG(2,q) having non-empty intersection with each secant of . As the number of
B ⊂ \K K
secants of is k(k 1)/2, the size of is at least k 1. If this lower bound is attained,
K − B −
is said to be of minimum size. Also, is linear if it is contained in a line.
B B
Arcs in PG(2,q) admitting a linear blocking set of minimum size of their secants are
called hyperfocused arcs. As mentioned in the Introduction, hyperfocused arcs exist only
in PG(2,q) for q even. Therefore in the whole paper we assume q = 2r.
Throughout, we fix the following notation. Let (X ,X ,X ) be homogeneous coordinates
1 2 3
for points in PG(2,q), and let ℓ be the line of equation X = 0. Given a pair A = (a,b)
3
in F F , denote A the point∞in PG(2,q) with coordinates (a,b,1), and A the point
q q
× ∞
(a,b,0). Also, let ϕ be the projectivity
A
ϕ : (X ,X ,X ) (X +a X ,X +a X ,X ).
A 1 2 3 1 1 3 2 2 3 3
7→
Clearly, ϕ is an elation with axis ℓ , and conversely for any non-trivial elation ϕ with
A
axis ℓ there exists A F F , A =∞ (0,0), such that ϕ = ϕ .
q q A
∞ ∈ × 6
Given an additive subgroup G of F F , let (P) be the orbit of the point P
q q G
× K ∈
PG(2,q) ℓ under the action of the group
\ ∞
T := ϕ A G .
G A
{ | ∈ }
Clearly, any two orbits (P) and (Q) with P, Q PG(2,q) ℓ are projectively
G G
K K ∈ \ ∞
equivalent. For brevity, write for (O), where O = (0,0,1). Note that
G G
K K
:= A A G .
G
K { | ∈ }
A k-arc in PG(2,q) coinciding with (P) for some additive subgroup G F F and
G q q
K ⊂ ×
some P PG(2,q) ℓ will be called a translation arc.
∈ \ ∞
HYPERFOCUSED ARCS 3
3. Translation arcs
The following proposition shows that any translation arc is a hyperfocused arc.
Proposition 3.1. Let be a translation arc. Then there exists a blocking set of the
K
secants of of minimum size which is contained in ℓ .
K ∞
Proof. Let G be an additive subgroup of F F such that is projectively equivalent
q q
× K
to . To prove the assertion, it is enough to show that every secant of meets ℓ in
G
K K ∞
a point C for some C G (0,0) . For A,B G, A = B, let l be the secant of
AB
∞ ∈ \{ } ∈ 6 K
passing through A and B. The intersection point of l and ℓ is (A+B) . Then the
AB
claim is proved, as A+B is a non-zero element of G. ∞ ∞ (cid:3)
According to Proposition 3.1 groups G in both Examples 3.2 and 3.3 provide examples of
translation arcs .
G
K
Example 3.2. (see [7]) For any additive subgroup H of F , let G = (α,α2) α H .
q
{ | ∈ }
Example 3.3. For H any additive subgroup of F and i any positive integer with (i,r) =
q
1, let G = (α,α2i) α H . Notethatthearc iscontained ina translationhyperoval
G
{ | ∈ } K
(see [8, Ch. 8]).
The following result shows that any translation k-arc is either complete in PG(2,q) ℓ
\ ∞
(i.e. it is not contained in any (k + 1)-arc PG(2,q) ℓ ), or it is contained in a
′
K ⊂ \ ∞
translation 2k-arc.
Proposition 3.4. Let be a translation k-arc in PG(2,q). Assume that there exists a
G
K
point A PG(2,q) belonging to no secant of . Then the set := ϕ ( ) is a
G ′ G A G
∈ K K K ∪ K
translation 2k-arc.
Proof. Assume that A , A and A are three collinear points in . Clearly neither
1 2 3 ′ G
K K
nor ϕ ( ) can contain all of such points. Also, as ϕ is an involution we may assume
A G A
K
A ,A , A ϕ ( ). Note that the elation ϕ := ϕ acts on both and
1 2 ∈ KG 3 ∈ A KG A+A3 KG
ϕ ( ). Then, as ϕ(A ) = A, the secant of through ϕ(A ) and ϕ(A ) contains A,
A G 3 G 1 2
K K
which is a contradiction. Hence is a 2k-arc. It is actually a translation arc because
′
K
′ = G′, where G′ = G (G+A).
K K ∪
(cid:3)
Example 3.5. For q a square, let η F F such that η2 F . Choose b F ,
q √q √q √q
∈ \ ∈ ∈
b = 1, and let G = (α,α2) α F , A = (η,bη2). Then by Proposition 3.4 the
√q
6 { | ∈ }
point set := ϕ ( ) is a translation (2√q)-arc. Note that half of the points
′ G A G
K K ∪ K
of are contained in the conic of equation X X = X2, the other half in the conic
′ 2 3 1
X XK = X2 +(b+1)η2X2.
2 3 1 3
Example 3.5 provides hyperfocused arcs which are not contained in any regular hyperoval.
Actually, the existence of hyperfocused arcs which are neither contained in any hyperoval
nor in any proper subplane of PG(2,q) can be proved. The proof of such a result needs
the following two lemmas.
4 GIULIETTI ANDMONTANUCCI
Lemma 3.6. Let be a translation q-arc containing both points (0,0,1) and (1,1,1).
Then there exist α,Kβ F and a positive integer i with (i,r) = 1, such that
q
∈
2i 2i
= (x,y,1) αx+(α+1)y +βx +(β +1)y = 0 .
K { | }
Proof. Let ψ be the linear collineation
αγ
ψ : (X ,X ,X ) (αX +(α+1)X ,γX +(γ +1)X ,X ),
α,γ 1 2 3 1 2 1 2 3
7→
with α,γ F , and let = ψ ( ). Choose α and γ in such a way that the two points
q ′ α,γ
∈ K K
on ℓ which belong to no secant of are (1,0,0) and (0,1,0). Note that contains
′ ′
(0,0,∞1) and (1,1,1). Also, for eachKt F there exists exactly one point PK of on
q t ′
the line X = tX . Let F be the funct∈ion on F such that P = (F(t),t,1). As Kis a
2 3 q t ′
translation arc containing (0,0,1), the set (F(t),t) t F is an additive subgrKoup of
q
F F . This implies F(s+t) = F(s)+F(t{) for any|s,t∈ F}. Theorem 8.41 in [8] yields
q q q
F(t×) = t2i for some i with (i,r) = 1, that is ∈
2i
= (x,y,1) x = y ,
′
K { | }
whence
2i
= (x,y,1) (αx+(α+1)y) = (γx+(γ +1)y) .
K { | }
Then the assertion follows by letting β = γ2i. (cid:3)
Lemma 3.7. Assume that r has a proper divisor s > 2, and let q = 2s. Let =
′ G
with G = {(a,a2) | a ∈ Fq′}. Then there exist at most rs translation q-arcs contaiKning KK.
Proof. Let be any translation arc of size q containing . Then by Lemma 3.6 there
exist α,β IF with α = β, and a positive integer i with (Ki,r) = 1, such that
q
∈ 6
2 2i 2i+1
αa+(α+1)a +βa +(β +1)a = 0,
for any a Fq′. This means that the polynomial g(T) := αT +(α+1)T2 +βT2i +(β +
1)T2i+1 m∈ust be divisible by Tq′ + T. If 2i+1 < q this can only happen for g(T) 0,
′
that is i = 1, β = 1, α = 0. If 2i+1 = q , that is i = s 1, then α = 1, β =≡ 0.
′
Finally, if 2i+1 > q , then also 2i > q as (i,r) = 1. Write i =−us + v with u,v integers
′ ′
with 0 v < s. Then 2i = (q 1)2u2v + 2v, and g(T) mod Tq′ +T is the polynomial
′
αT +(α≤+1)T2+βT2v +(β+1)T−2v+1, which has to be the zero polynomial. This implies
β = 1, α = 0, i s+1,2s+1,...,(r 1)s+1 . Note that the arc defined by i = s 1,
∈ { s − } −
α = 1, β = 0 coincides with that defined by i = (r 1)s+ 1, α = 0, β = 1. Then the
s − (cid:3)
assertion follows.
Now we are in a position to prove the following theorem.
Theorem 3.8. Let q = 2r be such that there r admits a proper divisor s > 2. Then there
exists a translation arc in PG(2,q) such that
K
(a) every point in PG(2,q) ℓ belongs to some secant of ;
(b) is not contained in an\y ∞hyperoval; K
K
(c) is not contained in any proper subplane.
K
HYPERFOCUSED ARCS 5
Proof. Let be as in Lemma 3.7, and let ,..., be the translation q-arcs containing
G 1 h
K I I
. Note that h r by Lemma 3.7. As there are exactly q (q 1)/2 secants of
KG ≤ s ′ ′ −
, the number of points in PG(2,q) ℓ contained in no secant of is at least
G G
Kq2 q(q 2 q )/2 = q(2r 22s 1 + 2s 1)\. O∞n the other hand, the numbKer of points in
′ ′ − −
− i−s at most qr. It−is straightforward to check that 2r 22s 1+2s 1 > r. Hence,
∪i=1,...,hIi s − − − s
there exists a point A PG(2,q) ℓ which is contained neither in a nor in a secant
1 i
∈ \ ∞ I
of . Define G = G + A and = . If every point in PG(2,q) ℓ belongs
KG 1 1 K1 KG1 \ ∞
to some secant of , let := . Otherwise choose a point A not belonging to any
1 1 2
K K K
secant of and let G = G +A , = . Repeat the process until the arc has
K1 2 1 2 K2 KG2 Ki
the property that every point in PG(2,q) ℓ belongs to some secant of , and define
i
\ ∞ K
= . Clearly, (a) is fulfilled by construction. Assume now that is contained in a
i
K K K
hyperoval . By (a), coincides with the points of not on ℓ , that is is one of the
′ ′
I K I ∞ K
translation q-arcs containing . But this is impossible as is not contained in any
G 1 i
K K I
by construction. Finally, (c) holds when s is chosen to be the maximum proper divisor
of r. In fact, in this case the maximum order of a subplane of PG(2,q) is 2s, whereas
# 2s+1 > 2s +2.
K ≥
(cid:3)
Theorem 3.8 suggests that it might be hard to deal with the problem of characterizing
hyperfocused arcs.
4. Generalized hyperfocused arcs
In this section we consider generalized hyperfocused arcs, that is arcs admitting a non-
necessarily linear blocking set of minimum size. In [1] it is shown that an arc in PG(2,q),
q even, does not admit a non-linear blocking set of its secants of minimum size, provided
that it is contained in a conic. The following theorem proves that k-arcs admitting non-
linear blocking sets of size k 1 actually exist.
−
Theorem 4.1. Let be a translation k-arc, k 4, and let ϕ be a homology with axis ℓ
and centre not in K. If the set = ϕ( )≥is an arc, then there exists a non-linea∞r
′
K K K ∪ K
blocking set of the secants of of minimum size.
′
B K
Proof. Assume that (0,0,1) , and let = , with G an additive subgroup of
G
∈ K K K
F F . Let C be the centre of ϕ. Define as the subset of 2k 1 points PG(2,q) which
q q
× B −
comprises points A , together with the centres of the homologies ϕϕ , with A ranging
A
∞
over G (0,0) . Let l be any secant of . If both P and Q are either in or in ϕ( ),
PQ ′
\{ } K K K
then l meets in a point A , for some A G (0,0) . Now assume that P = A and
PQ
B ∞ ∈ \{ }
Q = ϕ(B) for some A,B G. Then l passes through the centre of ϕϕ . This proves
PQ A+B
∈
that is a blocking set of the secants of . As has size 2k 1 and is not contained in
′
B K B − (cid:3)
any line, the assertion is proved.
Example 4.2. Let = with G = (0,0),(0,1),(1,0),(1,1) . Consider the homology
G
K K { }
(4.1) ϕ : (X ,X ,X ) (λX +a X ,λX +a X ,X ),
1 2 3 1 1 3 2 2 3 3
7→
6 GIULIETTI ANDMONTANUCCI
with
λ F , λ = 0,1, a ,a F ;
q 1 2 q
• ∈ 6 ∈
a ,a ,a +a 0,1,λ,λ+1 = .
1 2 1 2
• { }∩{ } ∅
Then it is straightforward to check that = ϕ( ) is an arc. A non-linear blocking
′
K K∪ K
set of the secants of of minimum size is
′
B K
= (1,0,0),(0,1,0),(1,1,0),(a ,a ,1+λ),(a +λ,a ,1+λ),
1 2 1 2
B {
(a ,a +λ,1+λ),(a +λ,a +λ,1+λ) ,
1 2 1 2
}
which consists of the points of a subplane of PG(2,q) of order 2.
The following result shows that a non-linear blocking set of minimum size of the secants
of a k-arc cannot be an arc itself. Also, it will be useful for the classification of small
generalized hyperfocused arcs which will be given in next section.
Proposition 4.3. Let be a blocking set of minimum size of the secants of a k-arc in
B K
PG(2,q), q even. Then any three points in blocking the secants of a 3-arc contained in
B
are collinear.
K
Proof. This proof relies on the idea of Segre’s celebrated Lemma of Tangents [10]. Let
P , P and P be any three distinct points in . For each i 1,2,3 , let Q be
1 2 3 i
K ∈ { } ∈ B
collinear with P and P , where j,k 1,2,3 , j = i+1 (mod 3), k = i 1 (mod 3). It
j k
∈ { } −
has to be proved that Q , Q and Q are collinear. Assume without loss of generality that
1 2 3
P = (1,0,0), P = (0,1,0) and P = (0,0,1). For a point P distinct from P , i = 1,2,3,
1 2 3 i
let α1, α2, α3 be the elements of F such that
P P P q
X = α1X is the line through P and P,
• X3 = αP2X2 is the line through P1 and P,
• 1 P 3 2
X = α3X is the line through P and P.
• 2 P 1 3
It is straightforward to check that if P does not belong to the triangle with vertices P ,
1
P , P , then
2 3
1 2 3
(4.2) α α α = 1.
P P P
Now, consider the set of secants of passing through exactly one point among P , P
1 2
K
and P . Clearly, it coincides with the set which comprises the lines joining P , P and P
3 1 2 3
to any point of Q ,Q ,Q , together with the lines through P and Q , i = 1,2,3.
1 2 3 i i
B \ { }
Hence,
1 2 3 1 2 3 1 2 3
α α α = α α α α α α .
P P P Q1 Q2 Q3 Q Q Q
!
P∈K,PY6=P1,P2,P3 Q∈B,QY6=Q1,Q2,Q3
Then by (4.2), α1 α2 α3 = 1 holds. As q is even, this is equivalent to the collinearity
Q1 Q2 Q3
of Q , Q and Q and the assertion is proved. (cid:3)
1 2 3
HYPERFOCUSED ARCS 7
5. Classification of small generalized hyperfocused arcs
The aim of this section if to classify the small arcs admitting blocking sets of minimum
size for their secants. The linear case has already been settled in [7] and [5]. The main
result of the section is the following.
Theorem 5.1. Let be a k-arc in PG(2,q), q even, with k 10. If there exists a
K ≤
minimal non-linear blocking set of the secants of , then k = 8 and is projectively
K K
equivalent to the arc in Example 4.2.
′
K
The proof of this result relies on a connection between blocking sets of the secants of an
arc and 1-factorizations of complete graphs. For the sake of completeness, some basic
definitions from graph theory are reported.
Let K be the complete graph with 2n vertices. A 1-factor of K is a set of vertex
2n 2n
disjoint edges which cover the vertices of K . An edge disjoint set of 1-factors covering
2n
the edges of K is said to be a 1-factorization of K . The set of vertices of K will be
2n 2n 2n
denoted by V(K ).
2n
Definition 5.2. Let be a 1-factorization of K . An embedding of in PG(2,q) is
2n
F F
an injective map ψ : V(K ) PG(2,q) such that
2n
∪F →
i) for any i,j,k V(K ), the points ψ(i), ψ(j), ψ(k) are not collinear;
2n
∈
ii) for any F , the point ψ(F) is collinear with ψ(i) and ψ(j), for every edge
∈ F
(i,j) F.
∈
Given an embedding ψ of a 1-factorization of K in PG(2,q), the set ψ(V(K )) is
2n 2n
F
an arc, whereas ψ( ) is a blocking set of minimum size of the secant of such arcs. The
F
following equivalent formulation of Theorem 5.1 will be proved.
Theorem 5.3. Let ψ be an embedding of a 1-factorization of K in PG(2,q), q even,
2n
F
with 3 n 5. If the points ψ(F) F are not collinear, then n = 4 and
≤ ≤ { | ∈ F}
ψ(V(K )) is projectively equivalent to the arc in Example 4.2.
2n ′
K
Assume that V(K ) = 1,2,...,2n , n 3, and let = F ,F ,...,F be a
2n 1 2 2n 1
{ } ≥ F { − }
1-factorization of K . Let ψ be an embedding of in PG(2,q).
2n
F
5.1. Proof of Theorem 5.3 for n = 3. As all the 1-factorizations of the complete graph
with 6 vertices are isomorphic, we may assume that:
ψ(F ) is the common point of the lines ψ(1)ψ(2), ψ(3)ψ(4), ψ(5)ψ(6);
1
•
ψ(F ) is the common point of the lines ψ(1)ψ(3), ψ(2)ψ(5), ψ(4)ψ(6);
2
•
ψ(F ) is the common point of the lines ψ(1)ψ(4), ψ(2)ψ(6), ψ(3)ψ(5);
3
•
ψ(F ) is the common point of the lines ψ(1)ψ(5), ψ(2)ψ(4), ψ(3)ψ(6);
4
•
ψ(F ) is the common point of the lines ψ(1)ψ(6), ψ(2)ψ(3), ψ(4)ψ(5).
5
•
By Proposition 4.3 the following triples of points are collinear:
ψ(F ),ψ(F ),ψ(F ), ψ(F ),ψ(F ),ψ(F ), ψ(F ),ψ(F ),ψ(F ).
1 2 3 1 2 4 1 2 5
Then all points in ψ(F) F are collinear, which proves the assertion.
{ | ∈ F}
8 GIULIETTI ANDMONTANUCCI
5.2. Proof of Theorem 5.3 for n = 4. There are 6 non-isomorphic 1-factorizations of
K (see e.g. [2]). From the proof of Theorem 5.3 in [5], it follows that 4 of them cannot
8
be embedded in PG(2,q). We are left with the following two cases.
Case 1: = F ,...,F with
1 7
F { }
F = (8,1),(2,3),(4,5),(6,7) , F = (8,2),(1,3),(4,6),(5,7) ,
1 2
{ } { }
F = (8,3),(1,2),(4,7),(5,6) , F = (8,4),(1,5),(2,6),(3,7) ,
3 4
{ } { }
F = (8,5),(1,4),(2,7),(3,6) , F = (8,6),(1,7),(2,4),(3,5) ,
5 6
{ } { }
F = (8,7),(1,6),(2,5),(3,4) .
7
{ }
Assume without loss of generality that ψ(4) = (0,0,1), ψ(5) = (0,1,1), ψ(6) = (1,0,1),
ψ(7) = (1,1,1), that is ψ(4),ψ(5),ψ(6),ψ(7) coincides with , with G as in Example
G
{ } K
4.2. Then ψ(F ) = (0,1,0), ψ(F ) = (1,0,0) and ψ(F ) = (1,1,0). Now, note that by
1 2 3
Proposition 4.3 the following triples of points are collinear:
ψ(F ),ψ(F ),ψ(F ), ψ(F ),ψ(F ),ψ(F ), ψ(F ),ψ(F ),ψ(F ).
4 5 1 4 6 2 4 7 3
Hence, if ψ(F ) lies on ℓ , then the whole ψ(F) F is contained in a line. Now
4
∞ { | ∈ F}
assume that ψ(F ) / ℓ .
4
∈ ∞
Let ϕ be the linear collineation of PG(2,q) such that ϕ(ψ(4)) = ψ(8), ϕ(ψ(5)) = ψ(1),
ϕ(ψ(6)) = ψ(2) and ϕ(ψ(7)) = ψ(3). Clearly, ϕ fixes ψ(F ), ψ(F ), ψ(F ), and hence
1 2 3
ϕ is a central collineation with axis ℓ . The centre of ϕ is ψ(F ), which is assumed
4
not to belong to ℓ . Therefore, ϕ is a∞s in Equation (4.1) for some a ,a ,λ F . As
1 2 q
∞ ∈
ψ(V(K )) = ϕ( ) is an arc, it is straightforward to check that
8 G G
K ∪ K
λ F , λ = 0,1, a ,a F ;
q 1 2 q
• ∈ 6 ∈
a ,a ,a +a 0,1,λ,λ+1 = .
1 2 1 2
• { }∩{ } ∅
Then the assertion is proved.
Case 2: = F ,...,F with
1 7
F { }
F = (8,1),(2,3),(4,5),(6,7) , F = (8,2),(1,4),(3,6),(5,7) ,
1 2
{ } { }
F = (8,3),(1,6),(2,5),(4,7) , F = (8,4),(1,7),(2,6),(3,5) ,
3 4
{ } { }
F = (8,5),(1,2),(3,7),(4,6) , F = (8,6),(1,5),(2,7),(3,4) ,
5 6
{ } { }
F = (8,7),(1,3),(2,4),(5,6) .
7
{ }
By Proposition 4.3, any point ψ(F ) with 3 i 7 is collinear with ψ(F ) and ψ(F ).
i 1 2
≤ ≤
Then all points in ψ(F) F are collinear, which proves the assertion.
{ | ∈ F}
5.3. Proofof Theorem 5.3 for n = 5. Define 0 asthesetofalltriples F ,F ,F such
i j k
T { }
that (i,j) F , (i,k) F , (j,k) F , with i,j,Fk ranging over V(K ). By Proposition
k j i 10
4.3, for any∈ F ,F ,F ∈ 0 the p∈oints ψ(F ), ψ(F ) and ψ(F ) are collinear.
i j k i j k
{ } ∈ T
F
Now define recursively a set i, i 1, as follows: i contains all the joins of two
sets in i 1 sharing at least tTwFo ele≥ments of . CleaTrFly, for any i, the points
−
T F A ∈ T
ψ(F) FF are collinear. By the following lemma, all points in ψ(F)F F are
{ | ∈ A} { | ∈ F}
collinear, which completes the proof of Theorem 5.3.
Lemma 5.4. For any 1-factorization of K , there exists an integer i for which i
10
F T
contains . F
F
HYPERFOCUSED ARCS 9
The proof of Lemma 5.4 consists of a computer based investigation of all 396 non-
isomorphic 1-factorizations of K ([2]). For the details of the proof the reader is referred
10
to [13].
References
[1] A. Aguglia, G. Korchm´arosand A. Siciliano, Minimal coverings of all chords of a conic in PG(2,q),
q even, Simon Stevin, to appear.
[2] L.D.Andersen,Factorizations of graphs.InThe CRC handbook of combinatorial designs,CRCPress
Series on Discrete Mathematics and its Applications, Boca Raton, CA, (1996) pp. 653–667.
[3] A.BeutelspacherandF.Wettl, On2-levelsecretsharing,Des. Codes Cryptogr., Vol.3,No.2(1993)
pp. 127–134.
[4] A. Bichara and G. Korchm´aros,Note on (q+2)-sets in a Galois plane of order q. In Combinatorial
and Geometric Structures and their Applications, Vol. 14 of Ann. Discrete Math., pp. 117–122.
North-Holland, Amsterdam, 1982.
[5] W.E. Cherowitzo and L.D. Holder, Hyperfocused Arcs, Simon Stevin, Vol. 12, No. 5 (2005) pp.
685–696.
[6] D. Drake, Hyperovals in nets of small degree, J. Combin. Des., Vol. 10 (2002) pp. 322–334.
[7] D.DrakeandK.Keating,OvalsandhyperovalsinDesarguesiannets,Des. Codes Cryptogr., Vol. 31
(2004), pp. 195–212.
[8] J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford (1998).
[9] L.D.Holder,TheconstructionofGeometricThresholdSchemeswithProjectiveGeometry,Master’s
Thesis, University of Colorado at Denver, 1997.
[10] B. Segre, Ovals in a finite projective plane, Canad. J. Math, Vol. 7 (1955), pp. 414–416.
[11] G. Simmons, Sharply Focused Sets of Lines on a Conic in PG(2,q), Congr. Numer., Vol. 73 (1990)
pp. 181–204.
[12] F. Wettl, On the nuclei of a pointset of a finite projective plane, J. Geom., Vol. 30 (1985), pp.
157–163.
[13] www.dipmat.unipg.it/giuliet/triangles.tex
e
Dipartimento di Matematica Universita` di Perugia, 06123 Perugia, Italy
E-mail address: [email protected], [email protected]