arXiv subjects

Edyta Bartnicka

Publications and source records attributed to Edyta Bartnicka.

8 recordsLinked to original sources

arXiv2025

Tops of graphs of projective codes

Let $Γ_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. Denote by $Π[n,k]_q$ the subgraph of $Γ_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^Π_{k}$ of $Π[n,k]_q$ consisting of all $k$-dimensional projective codes contained in a fixed $(k+1)$-dimensional subspace of $V$. We show when and in how many lines of ${\mathcal G}_{k}(V)$ they are contained. Next we prove that $\langle U]^Π_{k}$ is a maximal clique of $Π[n,k]_q$ exactly if it is contained in at most one line of ${\mathcal G}_{k}(V)$.

math.CO↗

arXiv2025

Tops of graphs of non-degenerate linear codes

Let $Γ_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $Γ(n,k)_q$ with the vertex set ${\mathcal C}(n,k)_q$ consisting of all non-degenerate linear $[n, k]_q$ codes. %We assume that $1<k<n-1$. We study maximal cliques $\langle U]^{c}_{k}$ of $Γ(n,k)_q$, which are intersections of tops of $Γ_k(V)$ with ${\mathcal C}(n,k)_q$. We show when they are contained in a line of ${\mathcal G}_{k}(V)$ and then we prove that $\langle U]^{c}_{k}$ is a maximal clique of $Γ(n,k)_q$ when it is not contained in a line of ${\mathcal G}_{k}(V)$. Furthermore, we show that the automorphism group of the set of such maximal cliques is isomorphic with the automorphism group of $Γ(n,k+1)_{q}$.

math.CO↗

arXiv2024

Stars of graphs of projective codes

Let $Γ_k(V)$ be the Grassmann graph whose vertex set is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $Π(n,k)_q$ formed by projective codes. We show that there are precisely two types of maximal cliques in $Π(n,k)_q$: stars and tops. We give a complete description of stars, i.e., maximal cliques consisting of all $k$-dimensional projective codes containing a certain $(k-1)$-dimensional subspace of $V$.

math.CO↗

arXiv2021

Orbits of Free Cyclic Submodules over Rings of Lower Triangular Matrices

Given a ring $T_n, n\geqslant 2$, of lower triangular $n\times n$ matrices with entries from an arbitrary field $F$, a complete description is performed of the orbits of free cyclic submodules of $^2T_n$, under the action of the general linear group $GL_2(T_n)$. It is given its total number, which is equal to the Bell number $B_n$, and there are shown their representatives.

math.RA↗

arXiv2019

Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices

Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order $q$, with the `line at infinity' being represented by those free cyclic submodules of ${^2T_n(q)}$ that are generated by non-unimodular pairs. Our approach can straightforwardly be adjusted to address the case of arbitrary $n$.

math.RA↗

arXiv2018

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

math.CO↗

arXiv2017

The distant graph of the projective line over a finite ring with unity

We discuss the projective line $\mathbb{P}(R)$ over a finite associative ring with unity. $\mathbb{P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation "distant". We study the graph of this relation on $\mathbb{P}(R)$ and classify up to isomorphism all distant graphs $G(R, Δ)$ for rings $R$ up to order $p^5$, $p$ prime.

math.RA↗

arXiv2015

Free cyclic submodules in the context of the projective line

We discuss the free cyclic submodules over an associative ring $R$ with unity. Special attention is paid to those, which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular $3$x$3$ matrices over a field $F$ under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.

math.RA↗