Related papers: q-Analogs of Steiner Systems

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…

The $q$-analogs of basic designs are discussed. It is proved that the existence of any unknown Steiner structures, the $q$-analogs of Steiner systems, implies the existence of unknown Steiner systems. Optimal $q$-analogs covering designs…

Fix a prime power $q$ and parameters $1\leq t\leq k\leq n$, the corresponding Steiner system in the Grassmann scheme, or the $q$-Steiner system, is a collection $\mathfrak{B}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^n$ such that for…

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.

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…

Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A $q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of $k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of $V$ is contained in a…

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…

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,…

A mixed Steiner system MS$(t,k,Q)$ is a set (code) $C$ of words of weight $k$ over an alphabet $Q$, where not all coordinates of a word have the same alphabet size, each word of weight $t$, over $Q$, has distance $k-t$ from exactly one…

Two mappings in a finite field, the Frobenius mapping and the cyclic shift mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to form automorphisms groups in the Grassmanian and in its codes. These automorphisms are…

A $t$-$(n,k,\lambda)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\F_q$ are…

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner $k$-system (for $k \geq 2$) is a linear space such that each line has size exactly $k$. Clearly, as a two-sorted structure,…

The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the Grassmannian space can be used for error-correction in random network coding. On the…

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…

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 Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. K\"{o}tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network…

In this paper we present a graph theoretic construction of Steiner quadruple systems (SQS) admitting abelian groups as point-regular automorphism groups. The resulting SQS has an extra property which we call A-reversibility, where A is the…

A Steiner triple system is a set $S$ together with a collection $\mathcal{B}$ of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly one element of $\mathcal{B}$. It is well known that the class of finite Steiner…

Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid…

A Steiner triple system, STS$(v)$, is a family of $3$-subsets (blocks) of a set of $v$ elements such that any two elements occur together in precisely one block. A collection of triples consisting of two copies of each block of an STS is…