59 research outputs found
Blocking Sets in the complement of hyperplane arrangements in projective space
It is well know that the theory of minimal blocking sets is studied by
several author. Another theory which is also studied by a large number of
researchers is the theory of hyperplane arrangements. We can remark that the
affine space is the complement of the line at infinity in .
Then can be regarded as the complement of an hyperplane arrangement
in ! Therefore the study of blocking sets in the affine space
is simply the study of blocking sets in the complement of a finite
arrangement in . In this paper the author generalizes this remark
starting to study the problem of existence of blocking sets in the complement
of a given hyperplane arrangement in . As an example she solves the
problem for the case of braid arrangement. Moreover she poses significant
questions on this new and interesting problem
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
Remarks on low weight codewords of generalized affine and projective Reed-Muller codes
We propose new results on low weight codewords of affine and projective
generalized Reed-Muller codes. In the affine case we prove that if the size of
the working finite field is large compared to the degree of the code, the low
weight codewords are products of affine functions. Then in the general case we
study some types of codewords and prove that they cannot be second, thirds or
fourth weight depending on the hypothesis. In the projective case the second
distance of generalized Reed-Muller codes is estimated, namely a lower bound
and an upper bound of this weight are given.Comment: New version taking into account recent results from Elodie Leducq on
the characterization of the next-to-minimal codewords (cf. arXiv:1203.5244
On subspace designs
Guruswami and Xing introduced subspace designs in 2013 to give the first
construction of positive rate rank metric codes list-decodable beyond half the
distance. In this paper we provide bounds involving the parameters of a
subspace design, showing they are tight via explicit constructions. We point
out a connection with sum-rank metric codes, dealing with optimal codes and
minimal codes with respect to this metric. Applications to two-intersection
sets with respect to hyperplanes, two-weight codes, cutting blocking sets and
lossless dimension expanders are also provided
Sweeps, arrangements and signotopes
AbstractSweeping is an important algorithmic tool in geometry. In the first part of this paper we define sweeps of arrangements and use the “Sweeping Lemma” to show that Euclidean arrangements of pseudolines can be represented by wiring diagrams and zonotopal tilings. In the second part we introduce a further representation for Euclidean arrangements of pseudolines. This representation records an “orientation” for each triple of lines. It turns out that a “triple orientation” corresponds to an arrangement exactly if it obeys a generalized transitivity law. Moreover, the “triple orientations” carry a natural order relation which induces an order relation on arrangements. A closer look on the combinatorics behind this leads to a series of signotope orders closely related to higher Bruhat orders. We investigate the structure of higher Bruhat orders and give new purely combinatorial proofs for the main structural properties. Finally, we reconnect the combinatorics of the second part to geometry. In particular, we show that the maximum chains in the higher Bruhat orders correspond to sweeps
Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal
Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal. © 2021, The Author(s)
- …