Related papers: $q$-Analogs for Steiner Systems and Covering Desig…
Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog of a Steiner system (briefly, a $q$-Steiner system), denoted $S_q[t,k,n]$, is a set $S$ of $k$-dimensional subspaces of $\F_q^n$ such that each…
One of the most intriguing problems, in $q$-analogs of designs, is the existence question of an infinite family of $q$-analog of Steiner systems, known also as $q$-Steiner systems, (spreads not included) in general, and the existence…
A Steiner structure $\dS = \dS_q[t,k,n]$ is a set of $k$-dimensional subspaces of $\F_q^n$ such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one subspace of $\dS$. Steiner structures are the $q$-analogs of Steiner…
One of the most intriguing problems, in $q$-analogs of designs and codes, is the existence question of an infinite family of $q$-analog of Steiner systems (spreads not included) in general, and the existence question for the $q$-analog of…
A well known class of objects in combinatorial design theory are {group divisible designs}. Here, we introduce the $q$-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces,…
In this paper we give the first construction of a q-analog of a Steiner system. Using a computer search we found at least 26 q-Steiner Systems S_2[2,3,13] admitting the normalizer of a singer cycle as a group of automorphisms.
The $q$-Fano plane is the $q$-analog of the Steiner system $S(2,3,7)$. For any given prime power $q$ it is not known whether the $q$-Fano plane exists or not. We consider the structure of the $q$-Fano plane for any given $q$ and conclude…
In this paper we prove that rank metric codes with special properties imply the existence of $q$-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a $[2d,d,d]$ dually almost MRD code $C\leq…
A $q$-covering design $\mathbb{C}_q(n, k, r)$, $k \ge r$, is a collection $\mathcal X$ of $(k-1)$-spaces of $\mathrm{PG}(n-1, q)$ such that every $(r-1)$-space of $\mathrm{PG}(n-1, q)$ is contained in at least one element of $\mathcal X$ .…
The von Neumann type subsystems of $q$-deformed coherent states are considered. The completeness of such subsystems is proved.
Intersection numbers for subspace designs are introduced and $q$-analogs of the Mendelsohn and K\"ohler equations are given. As an application, we are able to determine the intersection structure of a putative $q$-analog of the Fano plane…
Nested Steiner quadruple systems are designs derived from Steiner quadruple systems (SQSs) by partitioning each block into pairs. A nested SQS is completely uniform if every possible pair appears with equal multiplicity, and completely…
The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and…
A $t\text{-}(n,K,\lambda;q)$ design, also called the $q$-analog of a $t$-wise balanced design, is a set ${\mathcal B}$ of subspaces with dimensions contained in $K$ of the $n$-dimensional vector space ${\mathbb F}_q^n$ over the finite field…
We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility…
We give a new proof of the existence of designs, which is much shorter and gives better bounds.
We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which…
A $P_q(t,k,n)$ $q$-packing design is a selection of $k$-subspaces of $\F_q^n$ such that each $t$-subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer-Mesner-method of prescribing a…
We show how the relation between $Q$-manifolds and Lie algebroids extends to ``higher'' or ``non-linear'' analogs of Lie algebroids. We study the identities satisfied by a new algebraic structure that arises as a replacement of operations…
The smallest set of admissible parameters of a $q$-analog of a Steiner system is $S_2[2,3,7]$. The existence of such a Steiner system -- known as a binary $q$-analog of the Fano plane -- is still open. In this article, the automorphism…