272,238 research outputs found
On the stability of small blocking sets
A stability theorem says that a nearly extremal object can be obtained from an extremal one by “small changes”. In this paper, we study the relation of sets having few 0-secants and blocking sets
Dominating sets in projective planes
We describe small dominating sets of the incidence graphs of finite
projective planes by establishing a stability result which shows that
dominating sets are strongly related to blocking and covering sets. Our main
result states that if a dominating set in a projective plane of order is
smaller than (i.e., twice the size of a Baer subplane), then
it contains either all but possibly one points of a line or all but possibly
one lines through a point. Furthermore, we completely characterize dominating
sets of size at most . In Desarguesian planes, we could rely on
strong stability results on blocking sets to show that if a dominating set is
sufficiently smaller than 3q, then it consists of the union of a blocking set
and a covering set apart from a few points and lines.Comment: 19 page
Small Minimal Blocking Sets inPG(2, q3)
AbstractWe extend the results of Polverino (1999, Discrete Math., 208/209, 469–476; 2000, Des. Codes Cryptogr., 20, 319–324) on small minimal blocking sets in PG(2,p3 ), p prime, p≥ 7, to small minimal blocking sets inPG (2, q3), q=ph, p prime, p≥ 7, with exponent e≥h. We characterize these blocking sets completely as being blocking sets of Rédei-type
Large weight code words in projective space codes
AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points
Dominating Sets in Projective Planes
We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result that shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order q>81 is smaller than 2q+2⌊q⌋+2 (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most 2q+q+1. In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines. © 2016 Wiley Periodicals, Inc
Blocking and double blocking sets in finite planes
In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q(2) of size q(2) + 2q + 2 admitting 1-,2-,3-,4-, (q + 1)- and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q(2) of size at most 4q(2)/3 + 5q/3, which is considerably smaller than 2q(2) - 1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q(2).
We also consider particular Andre planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results
On linear sets of minimum size
An -linear set of rank on a projective line
, containing at least one point of weight one, has size at
least (see [J. De Beule and G. Van De Voorde, The minimum size of a
linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124.]). The classical
example of such a set is given by a club. In this paper, we construct a broad
family of linear sets meeting this lower bound, where we are able to prescribe
the weight of the heaviest point to any value between and . Our
construction extends the known examples of linear sets of size in
constructed for [G. Bonoli and O. Polverino,
-Linear blocking sets in , Innov. Incidence
Geom. 2 (2005), 35--56.] and in [G. Lunardon and O. Polverino. Blocking
sets of size . J. Comb. Theory, Ser: A 90 (2000), 148-158.]. We
determine the weight distribution of the constructed linear sets and describe
them as the projection of a subgeometry. For small , we investigate whether
all linear sets of size arise from our construction.
Finally, we modify our construction to define linear sets of size
in . This leads to new
infinite families of small minimal blocking sets which are not of R\'edei type
- …