Related papers: Constructions of General Covering Designs

We present constructions and results about GDDs with two groups and block size 6. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from…

Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ and they are not subsets of each other. We refer…

Given the finite set of $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ points $G_{n_1,n_2,\ldots,n_k} \subset \mathbb{R}^k$ such that $n_k \geq \cdots \geq n_2 \geq n_1 \in \mathbb{Z}^+$, we introduce a new algorithm, called M$\Lambda$I, which…

Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved…

A PSCA$(v, t, \lambda)$ is a multiset of permutations of the $v$-element alphabet $\{0, \dots, v-1\}$ such that every sequence of $t$ distinct elements of the alphabet appears in the specified order in exactly $\lambda$ permutations. For $v…

Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions…

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…

In 1940 Fisher famously showed that if there exists a non-trivial $(v,k,\lambda)$-design then $\lambda(v-1) \geq k(k-1)$. Subsequently Bose gave an elegant alternative proof of Fisher's result. Here, we show that the idea behind Bose's…

A \emph{covering array} is an $N \times k$ array of elements from a $v$-ary alphabet such that every $N \times t$ subarray contains all $v^t$ tuples from the alphabet of size $t$ at least $\lambda$ times; this is denoted as $\CA_\lambda(N;…

A $2$-$(v,k,\lambda)$ design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group $G$ in such a way that its block set is contained in (or coincides with) the set of all the zero-sum $k$-subsets of…

A connected covering is a design system in which the corresponding {\em block graph} is connected. The minimum size of such coverings are called {\em connected coverings numbers}. In this paper, we present various formulas and bounds for…

A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which…

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 $t\text{-}(n,K,\lambda;q)$ 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 try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete…

We introduce and investigate binary $(k,k)$-designs -- combinatorial structures which are related to binary orthogonal arrays. We derive general linear programming bound and propose as a consequence a universal bound on the minimum possible…

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain…

A generalized balanced tournament design, or a GBTD(k, m) in short, is a (km, k, k-1)-BIBD defined on a km-set V . Its blocks can be arranged into an m\times(km-1) array in such a way that (1) every element of V is contained in exactly one…

If $G$ is a finite group and $k =q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a $2$-$(v,k,1)$-design ${\bf D}$ for which ${\rm Aut} {\bf D}\cong G$.

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur…