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A t (modp) result on weighted multiple (n − k)-blocking sets in PG(n,q)

2008
*
Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial
*

A 1-fold (

doi:10.2140/iig.2008.6.169
fatcat:6w4zzurmsbg6fa7bhus4izet4y
*n*− k)-*blocking**set*of*PG*(*n*,*q*) is called small when it has less than*3*(*q**n*−k + 1)/2 points. ... A minimal 1-fold (*n*− k)-*blocking**set**in**PG*(*n*,*q*),*q*= p h , p > 2 prime, h ≥ 1, of size less than*3*2 (*q**n*−k + 1) intersects every subspace*in*zero points or*in*1 (mod p) points. ...##
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The invariant factors of the incidence matrices of points and subspaces in PG(n,q) and AG(n,q)
[article]

2003
*
arXiv
*
pre-print

of AG(

arXiv:math/0312506v1
fatcat:bvhynw5wybbvzk7xhg2fwnwn6q
*n*,*q*) for all*n*, r, and arbitrary prime power*q*. ... We determine the Smith normal forms of the incidence matrices of points and projective (r-1)-dimensional subspaces of*PG*(*n*,*q*) and of the incidence matrices of points and r-dimensional affine subspaces ... Let L 1 be the*set*of projective points and let L r be the*set*of projective (r − 1)-spaces*in**PG*(*n*,*q*), and let d i and H be as above. ...##
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Classes and equivalence of linear sets in PG(1,q^n)
[article]

2016
*
arXiv
*
pre-print

The equivalence problem of F_q-linear

arXiv:1607.06962v1
fatcat:3zypvrxe4fcn7ggnqthk3oauqi
*sets*of rank*n*of*PG*(1,*q*^*n*) is investigated, also*in*terms of the associated variety, projecting configurations, F_q-linear*blocking**sets*of Rédei type and MRD-codes ...*blocking**sets**in**PG*(2,*q*4 ). ...*In*the recent years, starting from the paper [18] by Lunardon, linear*sets*have been used to construct or characterize various objects*in*finite geometry, such as*blocking**sets*and*multiple**blocking*...##
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Construction of (n,r)-arcs in PG(2,q)

2005
*
Innovations in Incidence Geometry Algebraic Topological and Combinatorial
*

We can improve the known lower bounds for

doi:10.2140/iig.2005.1.133
fatcat:kf7ojbnjzja6rmatfmgmhvjlqa
*q*= 11, 13, 16, 17, 19 and give the first example of a double*blocking**set*of size*n**in*P G(2, p) with*n*< 3p and p prime. ... We construct new (*n*, r)-arcs*in*P G(2,*q*) by prescribing a group of automorphisms and solving the resulting Diophantine linear system with lattice point enumeration. ...*N*. Daskalov and M. E. J. Contreras, who made available copies of their papers. ...##
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The unipotent modules of $${{\mathrm {GL}}}_{n}({{\mathbb {F}}}_{q})$$ GL n ( F q ) via tableaux

2017
*
Journal of Algebraic Combinatorics
*

Note that G acts on the

doi:10.1007/s10801-017-0766-2
fatcat:63nxlwyford6zjpff2gwwcst6a
*set*of F*n**q*-tableaux by left*multiplication*of the entries (considered as column vectors). ... Unless λ = (1*n*), the*set*HW contains*multiple*representatives of the same coset; for example, U λ hP − λ = U λ P − λ for all h ∈ H. ...##
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The invariant factors of the incidence matrices of points and subspaces in $\operatorname {PG}(n,q)$ and $\operatorname {AG}(n,q)$

2006
*
Transactions of the American Mathematical Society
*

of AG(

doi:10.1090/s0002-9947-06-03859-1
fatcat:ph6umkjsubbpdkp4b7ejkhqitq
*n*,*q*) for all*n*, r, and arbitrary prime power*q*. ... We determine the Smith normal forms of the incidence matrices of points and projective (r − 1)-dimensional subspaces of*PG*(*n*,*q*) and of the incidence matrices of points and r-dimensional affine subspaces ... Let L 1 be the*set*of projective points, let L r be the*set*of projective (r − 1)-spaces*in**PG*(*n*,*q*), and let d i and H be as above. ...##
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On the upper chromatic number and multiple blocking sets of PG(n,q)

2019
*
Journal of combinatorial designs (Print)
*

We investigate the upper chromatic number of the hypergraph formed by the points and the k-dimensional subspaces of

doi:10.1002/jcd.21686
fatcat:ygvi6whs55gqhfs2wlzatpfyhy
*n**q**PG*( , ); that is, the most number of colors that can be used to color the points ... Let be a weighted t-fold*n*k ( − )-*blocking**set**in**n**q**PG*( , ). ... An m-space is a subspace of*n**q**PG*( , ) of dimension m (*in*projective sense). A point*set**in**n**q**PG*( , ) is called a t-fold*n*k ( − )-*blocking**set*if every k-space intersects*in*at least t points. ...##
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On Absolute Points of Correlations of $\mathrm{PG}(2,q^n)$

2020
*
Electronic Journal of Combinatorics
*

Kestenband regarding what is known for the

doi:10.37236/8920
fatcat:5ushs4lxfzaltjz72aoov7dtge
*set*of the absolute points of correlations*in*$\mathrm{*PG*}(2,*q*^*n*)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{*q*^*n*}^*3*$. ...*In*this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{*q*^*n*}^*3*$. ... We will determine*in*the next sections the*set*Γ*in**PG*(1,*q**n*) and*in**PG*(2,*q**n*). ...##
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On the upper chromatic number and multiplte blocking sets of PG(n,q)
[article]

2019
*
arXiv
*
pre-print

Due to this relation with double

arXiv:1909.02867v1
fatcat:krsgluavfbduxomcur523xkvhm
*blocking**sets*, we also prove that for t≤*3*/8p+1, a small t-fold (weighted) (*n*-k)-*blocking**set*of*PG*(*n*,p), p prime, must contain the weighted sum of t not necessarily distinct ... We investigate the upper chromatic number of the hypergraph formed by the points and the k-dimensional subspaces of*PG*(*n*,*q*); that is, the most number of colors that can be used to color the points so that ... NN 114614 (*in*Hungary) and N1-0032 (*in*Slovenia). The second author was also supported by the János Bolyai Research Grant. ...##
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On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

2008
*
Designs, Codes and Cryptography
*

We link the codewords of small weight of this code to

doi:10.1007/s10623-008-9203-9
fatcat:qkjipzxtrrc5hhpa5zddv5oyv4
*blocking**sets*with respect to lines*in*P G(*n*,*q*) and we exclude all possible codewords arising from small linear*blocking**sets*. ...*In*this paper, we study the p-ary linear code C(P G(*n*,*q*)),*q*= p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space P G(*n*,*q*), and its dual ... Corollary*3*.*3*The only possible codewords c of C of weight*in*]θ*n*−1 , 2q*n*−1 [ are the scalar*multiples*of non-linear minimal*blocking**sets*, intersecting every line*in*1 (mod p) points. ...##
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Codes arising from incidence matrices of points and hyperplanes in PG(n,q)
[article]

2016
*
arXiv
*
pre-print

p prime, proving that they are the scalar

arXiv:1606.02222v1
fatcat:ujtwmncdofbjvaaa2ks6fu7lxi
*multiples*of the difference of the incidence vectors of two distinct hyperplanes of*PG*(*n*,*q*). ...*In*this paper we completely characterize the words with second minimum weight*in*the p-ary linear code generated by the rows of the incidence matrix of points and hyperplanes of*PG*(*n*,*q*), with*q*=p^h and ... [11, Corollary 1] Every (*n*− 1)−*blocking**set**in*P G(*n*,*q*), of size smaller than*q**n*−1 + θ*n*−1 , can be uniquely reduced to a minimal (*n*− 1)−*blocking**set*. ...##
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Nilpotent subgroups of GL(n, [open face Q])

2001
*
Glasgow Mathematical Journal
*

We compute the precise bounds, for every positive integer

doi:10.1017/s0017089501030099
fatcat:fuddkve43remzmsxzd2sz5iygi
*n*, for the nilpotency class of nilpotent subgroups of GLðn; QÞ and GLðn; ZÞ. ... Þ is Galois over*Q*,*Q*6Â6 is the skew group ring of À over K and Å < À has order*3*. Further G embeds into ðÅ:K Ã Þwr P. By 3.7 the group G has class at most 3r and here*n*¼ 6r, so that k*n**3*¼*3*þ1 . ... Þ C, where w is a primitive 9th root of 1 and let À be the Sylow*3*-subgroup (of order*3*) of GalðK=QÞ.*Set*W ¼ ðÀ:K Ã ÞwrP, where P is now a transitive*3*-subgroup of SymðrÞ. ...##
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On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual

2008
*
Finite Fields and Their Applications
*

*In*this paper, we study the p-ary linear code C k (

*n*,

*q*),

*q*= p h , p prime, h 1, generated by the incidence matrix of points and k-dimensional spaces

*in*

*PG*(

*n*,

*q*). ... For k

*n*/2, we link codewords of C k (

*n*,

*q*) \ C k (

*n*,

*q*) ⊥ of weight smaller than 2q k to k-

*blocking*

*sets*. ... Let K be a

*blocking*

*set*

*in*

*PG*(

*n*,

*q*),

*n*

*3*, with |K| 2q − 1. ...

##
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An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)
[article]

2012
*
arXiv
*
pre-print

11, and

arXiv:1201.3297v1
fatcat:rpy5wsjmfrd3fcqtit2xux5sxm
*in*the case*n*= 2, for*q*= p^*3*, p >= 7 ([4],[5],[7],[8]). ...*In*par- ticular, for the code Cn-1(*n*,*q*) of points and hyperplanes of*PG*(*n*,*q*), we exclude all codewords*in*Cn-1(*n*,*q*) with weight*in*the open interval ]*q*^*n*-1/*q*-1, 2q^*n*-1[. ... Let Y be a linear small minimal (*n*− k)-*blocking**set**in**PG*(*n*,*q*). ...##
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The 2-Blocking Number and the Upper Chromatic Number ofPG(2,q)

2013
*
Journal of combinatorial designs (Print)
*

A 2-fold

doi:10.1002/jcd.21347
fatcat:fkalrojogjcapn246msvlsy7vq
*blocking**set*(double*blocking**set*)*in*a finite projective plane Π is a*set*of points, intersecting every line*in*at least two points. ... The minimum number of points*in*a double*blocking**set*of Π is denoted by τ 2 (Π). Let*PG*(2,*q*) be the Desarguesian projective plane over GF(*q*), the finite field of*q*elements. ... It is well-known that τ 2 (*PG*(2,*q*)) ≤ 2(*q*+ √*q*+ 1) if*q*is a square.*In*Section 2 we study*multiple**blocking**sets*. ...
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