arXiv subjects

Papa Sissokho

Publications and source records attributed to Papa Sissokho.

6 recordsLinked to original sources

arXiv2026

A note on Tight Irreducible Affine Spreads

Let ${\mathbb F}_q^n$ denote the vector space of dimension $n$ over ${\mathbb F}_q$ and AG$(n,q)$ denote the corresponding affine space. An $\textit{affine vector space partition}$ of AG$(n,q)$ is a collection ${\mathcal P}$ of affine subspaces that partition the points of AG$(n,q)$. If all subspaces in ${\mathcal P}$ have the same dimension $d$, then ${\mathcal P}$ is called an $\textit{affine $d$-spread}$. We say that an affine partition ${\mathcal P}$ is $\textit{completely tight}$ if for any pair $C,C'\in{\mathcal P}$ with $C=v+S$, $C'=v'+S'$, where $v,v'\in{\rm AG}(n,q)$ and $S\neq S'$ are linear subspaces of ${\mathbb F}_q^n$, we have $ S\cap S'=\{{\bf 0}\}$. An affine partition ${\mathcal P}$ is said to be $\textit{irreducible}$ if there is no subset ${\mathcal P}'\subset {\mathcal P}$ such that $1<|{\mathcal P}'|<|{\mathcal P}|$ and the union of all subspaces in ${\mathcal P}'$ is a subspace of AG$(n,q)$. For all $d \geq 1$ and $n>2d$, we construct a completely tight irreducible affine $d$-spread of AG$(n,q)$. This also settles a recent conjecture of Bamberg et al. on the existence of tight irreducible affine $d$-spreads.

math.CO↗

arXiv2017

The maximum size of a partial spread in a finite projective space

Let $n$ and $t$ be positive integers with $t (q^r-1)/(q-1)$, then the maximum size, i.e., cardinality, of a partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is $(q^n-q^{t+r})/(q^t-1)+1$. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for $r\in\{0,1\}$ and for $r=q=2$.

math.CO↗

arXiv2017

The maximum size of a partial spread II: Upper bounds

Let $n$ and $t$ be positive integers with $t 2$. The exact value of the maximum size partial $(t-1)$-spread has been recently determined for $t>θ_r$ by the authors of this paper (see Năstase-Sissokho [21]).

math.CO↗

arXiv2014

Level Matrices

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0 \ell$, any $m\times n$ level matrix with entries in $\{0,\ldots,k\}$ is reducible. It is known that $\ell(2,k)=2k-1$. In this paper, we establish the existence of $\ell(n,k)$ for $n\geq 3$ by giving upper and lower bounds for it. We then apply this result to bound the number of certain types of vector space multipartitions.

math.CO↗

arXiv2012

A zero-sum theorem over Z

A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\ell(k)$ be the smallest integer $\ell$ such that any zero-sum sequence with elements from $[-k,k]$ and length greater than $\ell$ contains a proper nonempty zero-sum subsequence. In this paper, we prove a more general result which implies that $\ell(k)=2k-1$ for $k>1$.

math.CO↗

arXiv2011

Extremal sizes of subspace partitions

A subspace partition $Π$ of $V=V(n,q)$ is a collection of subspaces of $V$ such that each 1-dimensional subspace of $V$ is in exactly one subspace of $Π$. The size of $Π$ is the number of its subspaces. Let $σ_q(n,t)$ denote the minimum size of a subspace partition of $V$ in which the largest subspace has dimension $t$, and let $ρ_q(n,t)$ denote the maximum size of a subspace partition of $V$ in which the smallest subspace has dimension $t$. In this paper, we determine the values of $σ_q(n,t)$ and $ρ_q(n,t)$ for all positive integers $n$ and $t$. Furthermore, we prove that if $n\geq 2t$, then the minimum size of a maximal partial $t$-spread in $V(n+t-1,q)$ is $σ_q(n,t)$.

math.CO↗