Generalising Fisher's inequality to coverings and packings

TL;DR

This paper extends Fisher's inequality to derive new bounds for the size of coverings and packings in combinatorial design theory, broadening the understanding of their structural limitations.

Contribution

It generalizes Bose's proof technique to establish novel bounds on the number of blocks in coverings and packings with specific parameter constraints.

Findings

New bounds on the number of blocks in coverings and packings

Generalization of Bose's proof method

Enhanced understanding of combinatorial design limitations

Abstract

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 proof can be generalised to obtain new bounds on the number of blocks in $(v,k,\lambda)$-coverings and -packings with $\lambda(v-1)<k(k-1)$.