Leo Storme
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Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$ Open
We investigate Cameron-Liebler sets of planes in the Klein quadric $Q^+(5,q)$ in PG$(5,q)$. We prove that there are many examples of such Cameron-Liebler sets of planes in the Klein quadric. More specifically, we provide an incomplete list…
The minimum weight of the code of intersecting lines in PG(3,q) Open
We characterise the minimum weight codewords of the p-ary linear code of intersecting lines in PG(3,q), q = p^h, q ≥ 19, p prime, h ≥ 1. If q is even, the minimum weight equals q³ + q² + q + 1. If q is odd, the minimum weight equals q³ + 2…
The minimum weight of the code of intersecting lines in ${\rm PG}(3,q)$ Open
We characterise the minimum weight codewords of the $p$-ary linear code of intersecting lines in ${\rm PG}(3,q)$, $q=p^h$, $q\geq19$, $p$ prime, $h\geq 1$. If $q$ is even, the minimum weight equals $q^3+q^2+q+1$. If $q$ is odd, the minimum…
View article: On two non-existence results for Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$
On two non-existence results for Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$ Open
This paper focuses on non-existence results for Cameron-Liebler $k$-sets. A Cameron-Liebler $k$-set is a collection of $k$-spaces in $\mathrm{PG}(n,q)$ or $\mathrm{AG}(n,q)$ admitting a certain parameter $x$, which is dependent on the size…
Maximal Sets of $k$-Spaces Pairwise Intersecting in at Least a $(k-2)$-Space Open
In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of t…
Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ Open
We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatoric…
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Finite geometries: pure mathematics close to applications Open
The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studi…
Maximal sets of $k$-spaces pairwise intersecting in at least a $(k-2)$-space Open
In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of t…
$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $ Open
In this paper we construct 2-PD-sets of 16 elements for codes from the Desarguesian projective planes PG(2, q), where q = 2(h) and 5 <= h <= 9. We also construct 3-PD-sets of 75 elements for the code from the Desarguesian projective plane …
Cameron-Liebler $k$-sets in $\text{AG}(n,q)$ Open
We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraic combinatorics,…
Optimal subspace codes in Open
We investigate subspace codes whose codewords are subspaces of PG(4, q) having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that Aq (5, 3) = 2(q(3)+1) 1).
On the independence number of graphs related to a polarity Open
We investigate the independence number of two graphs constructed from a polarity of . For the first graph under consideration, the Erdős‐Rényi graph , we provide an improvement on the known lower bounds on its independence number. In the s…
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of Combinatorial Designs; 24 issues total, 2018): Institutional subscription prices for 2018 are: Print & Online: US$5763 (US), US$6322 (Rest of World), €4080 (Europe), £3229 (UK).Prices are exclusive of tax.Asia-Pacific GST, Canadian GST …
A Characterization of Hermitian Varieties as Codewords Open
It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $\mathrm{PG}(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of point…
Optimal subspace codes in ${\rm PG}(4,q)$ Open
We investigate subspace codes whose codewords are subspaces of ${\rm PG}(4,q)$ having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that ${\cal A}_q(5,3) = 2(q^3+1)$.
A characterization of Hermitian varieties as codewords Open
It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces PG(r, q(2)). In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of PG(r…
On the independence number of graphs related to a polarity Open
We investigate the independence number of two graphs constructed from a polarity of $\mathrm{PG}(2,q)$. For the first graph under consideration, the Erdős-Rényi graph $ER_q$, we provide an improvement on the known lower bounds on its indep…
On primitive constant dimension codes and a geometrical sunflower bound Open
In this paper we study subspace codes with constant intersection dimension (SCIDs). We investigate the largest possible dimension spanned by such a code that can yield non-sunflower codes, and classify the examples attaining equality in th…
Tight sets in finite classical polar spaces Open
We show that every i -tight set in the Hermitian variety H (2 r + 1, q ) is a union of pairwise disjoint (2 r + 1)-dimensional Baer subgeometries PG ( 2 r + 1 , q ) $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H (2 r + 1, q ), if q …
Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes Open
We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in P-n(F-q) of small degree d, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have app…