New lower bounds for the number of blocks in balanced incomplete block designs

TL;DR

This paper establishes a new lower bound on the number of blocks in balanced incomplete block designs (BIBDs), which improves upon previous bounds for many cases, advancing the theoretical understanding of BIBDs.

Contribution

It introduces a novel lower bound for the number of blocks in BIBDs that surpasses existing bounds for a wide range of designs.

Findings

New lower bound for BIBDs established

Bound is tighter than Bose and Kageyama bounds in many cases

Advances theoretical limits in combinatorial design theory

Abstract

Bose proved the inequality $b\geq v+r-1$ for resolvable balanced incomplete block designs (RBIBDs) and Kageyama improved it for RBIBDs which are not affine resolvable. In this note we prove a new lower bound on the number of blocks $b$ that holds for all BIBDs. We further prove that for a significantly large number of BIBDs our bound is tighter than the bounds given by the inequalities of Bose and Kageyama.