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

Sandy Ferret, Leo Storme, Péter Sziklai, Zsuzsa Weiner
2008 Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial  
A 1-fold (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.  ... 
doi:10.2140/iig.2008.6.169 fatcat:6w4zzurmsbg6fa7bhus4izet4y

The invariant factors of the incidence matrices of points and subspaces in PG(n,q) and AG(n,q) [article]

David B. Chandler, Peter Sin, Qing Xiang
2003 arXiv   pre-print
of AG(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.  ... 
arXiv:math/0312506v1 fatcat:bvhynw5wybbvzk7xhg2fwnwn6q

Classes and equivalence of linear sets in PG(1,q^n) [article]

Bence Csajbók, Giuseppe Marino, Olga Polverino
2016 arXiv   pre-print
The equivalence problem of F_q-linear 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  ... 
arXiv:1607.06962v1 fatcat:3zypvrxe4fcn7ggnqthk3oauqi

Construction of (n,r)-arcs in PG(2,q)

Michael Braun, Axel Kohnert, Alfred Wassermann
2005 Innovations in Incidence Geometry Algebraic Topological and Combinatorial  
We can improve the known lower bounds for 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.  ... 
doi:10.2140/iig.2005.1.133 fatcat:kf7ojbnjzja6rmatfmgmhvjlqa

The unipotent modules of $${{\mathrm {GL}}}_{n}({{\mathbb {F}}}_{q})$$ GL n ( F q ) via tableaux

Scott Andrews
2017 Journal of Algebraic Combinatorics  
Note that G acts on the 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.  ... 
doi:10.1007/s10801-017-0766-2 fatcat:63nxlwyford6zjpff2gwwcst6a

The invariant factors of the incidence matrices of points and subspaces in $\operatorname {PG}(n,q)$ and $\operatorname {AG}(n,q)$

David B. Chandler, Peter Sin, Qing Xiang
2006 Transactions of the American Mathematical Society  
of AG(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.  ... 
doi:10.1090/s0002-9947-06-03859-1 fatcat:ph6umkjsubbpdkp4b7ejkhqitq

On the upper chromatic number and multiple blocking sets of PG(n,q)

Zoltán L. Blázsik, Tamás Héger, Tamás Szőnyi
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 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.  ... 
doi:10.1002/jcd.21686 fatcat:ygvi6whs55gqhfs2wlzatpfyhy

On Absolute Points of Correlations of $\mathrm{PG}(2,q^n)$

Jozefien D'haeseleer, Nicola Durante
2020 Electronic Journal of Combinatorics  
Kestenband regarding what is known for the 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 ).  ... 
doi:10.37236/8920 fatcat:5ushs4lxfzaltjz72aoov7dtge

On the upper chromatic number and multiplte blocking sets of PG(n,q) [article]

Zoltán L. Blázsik, Tamás Héger, Tamás Szőnyi
2019 arXiv   pre-print
Due to this relation with double 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.  ... 
arXiv:1909.02867v1 fatcat:krsgluavfbduxomcur523xkvhm

On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

Michel Lavrauw, Leo Storme, Geertrui Van de Voorde
2008 Designs, Codes and Cryptography  
We link the codewords of small weight of this code to 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 inn−1 , 2q n−1 [ are the scalar multiples of non-linear minimal blocking sets, intersecting every line in 1 (mod p) points.  ... 
doi:10.1007/s10623-008-9203-9 fatcat:qkjipzxtrrc5hhpa5zddv5oyv4

Codes arising from incidence matrices of points and hyperplanes in PG(n,q) [article]

Olga Polverino, Ferdinando Zullo
2016 arXiv   pre-print
p prime, proving that they are the scalar 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.  ... 
arXiv:1606.02222v1 fatcat:ujtwmncdofbjvaaa2ks6fu7lxi

Nilpotent subgroups of GL(n, [open face Q])

B. A. F. Wehrfritz
2001 Glasgow Mathematical Journal  
We compute the precise bounds, for every positive integer 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 à 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 à ÞwrP, where P is now a transitive 3-subgroup of SymðrÞ.  ... 
doi:10.1017/s0017089501030099 fatcat:fuddkve43remzmsxzd2sz5iygi

On the code generated by the incidence matrix of points and k-spaces in PG(n,q) and its dual

M. Lavrauw, L. Storme, G. Van de Voorde
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.  ... 
doi:10.1016/j.ffa.2008.06.002 fatcat:hpifrjnznna5nmdvv5fyle3wcq

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

Michel Lavrauw, Leo Storme, Peter Sziklai, Geertrui Van de Voorde
2012 arXiv   pre-print
11, and 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).  ... 
arXiv:1201.3297v1 fatcat:rpy5wsjmfrd3fcqtit2xux5sxm

The 2-Blocking Number and the Upper Chromatic Number ofPG(2,q)

Gábor Bacsó, Tamás Héger, Tamás Szőnyi
2013 Journal of combinatorial designs (Print)  
A 2-fold 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.  ... 
doi:10.1002/jcd.21347 fatcat:fkalrojogjcapn246msvlsy7vq
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