Constructions of General Covering Designs

TL;DR

This paper introduces new methods for constructing general covering designs, expanding existing techniques and providing improved upper bounds on the minimum size of these combinatorial structures.

Contribution

The paper presents novel constructions for general covering designs and generalizes existing methods, leading to tighter bounds on their minimal sizes.

Findings

New constructions for general covering designs

Generalization of existing design construction methods

Improved upper bounds on minimum design size

Abstract

Given five positive integers $v, m,k,\lambda$ and $t$ where $v \geq k \geq t$ and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and $\mathcal{B}$ a multiset of $k$-subsets of $X$ (called blocks) such that every $m$-subset of $X$ intersects (is covered by) at least $\lambda$ members of $\mathcal{B}$ in at least $t$ points. In this article we present new constructions for general covering designs and we generalize some others. By means of these constructions we will be able to obtain some new upper bounds on the minimum size of such designs.