The quadric Veronesean
V
2
4
in PG
(
5
,
q
)
is characterized as a
(
q
2
+
q
+
1
)
-cap of class
0
,
1
,
2
,
3
,
q
+
1
2
and type
(
1
,
q
+
1
,
2
q
+
1
)
4
of PG
(
5
,
q
)
by Ferri (
q odd) and ...Thas and Van Maldeghem (
q even). In this note we generalize this result slightly by proving that for a
(
q
2
+
q
+
1
)
-cap of class
0
,
1
,
2
,
3
,
q
+
1
2
and type
(
1
,
m
,
n
)
4
of PG
(
5
,
q
)
, the parameters
m and
n are uniquely determined and equal
q
+
1
and
2
q
+
1
, respectively.
Bruen proved that if
A is a set of points in AG(
n,
q) which intersects every hyperplane in at least
t points, then |
A|⩾(
n+
t−1)(
q−1)+1, leaving as an open question how good such bound is. Here we ...prove that, up to a trivial case, if
t>((
n−1)(
q−1)+1)/2, then Bruen's bound can be improved. If
t is equal to the integer part of ((
n−1)(
q−1)+1)/2, then there are some examples which attain such a lower bound. Somehow, this suggests the following combinatorial characterization: if a set
S of points in PG(3,
q) meets every affine plane in at least
q−1 points and is of minimum size with respect to this property, then
S is a hyperbolic quadric.
Partial linear complexes of PG(3, q) Metsch, K.; Storme, L.
Discrete mathematics,
08/2002, Letnik:
255, Številka:
1
Journal Article, Conference Proceeding
Recenzirano
Odprti dostop
In this article, it is shown that the largest partial linear complex of PG(3,
q) that is not contained in a linear complex has
q
3+3 lines. We also show that such an inextendable partial linear ...complex of size
q
3+3 is projectively unique except in the case
q=3.
Let
Y⊂
P
n
be a cubic hypersurface defined over GF(
q). Here, we study the Finite Field Nullstellensatz of order
q/3 for the set
Y(
q) of its GF(
q)-points, the existence of linear subspaces of PG(
...n,
q) contained in
Y(
q) and the possibility to join any two points of
Y(
q) by the union of two lines of PG(
n,
q) entirely contained in
Y(
q). We also study the existence of linear subspaces defined over GF(
q) for the intersection of
Y with
s quadrics and for quartic hypersurfaces.
We consider the Radon transform along lines in an n-dimensional vector space over the two-element field. It is well known that this transform is injective and highly overdetermined. We classify the ...minimal collections of lines for which the restricted Radon transform is also injective. This is an instance of I.M. Gelfand’s admissibility problem. The solution is in stark contrast to the more uniform cases of the affine hyperplane transform and the projective line transform, which are addressed in other papers (Feldman and Grinberg in Admissible Complexes for the Projective X-Ray Transform over a Finite Field, preprint, 2012; Grinberg in J. Comb. Theory, Ser. A 53:316–320, 1990). The presentation here is intended to be widely accessible, requiring minimum background.
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Provider: - Institution: - Data provided by Europeana Collections- All metadata published by Europeana are available free of restriction under the Creative Commons CC0 1.0 Universal Public Domain ...Dedication. However, Europeana requests that you actively acknowledge and give attribution to all metadata sources including Europeana
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