On the Inequality for BIBDs with Special Parameters

Abstract

For a $\mu$-resolvable Balanced Incomplete Block Design (BIBD) with parameters $v, b = mt, r = \mu t, k$ and $\lambda$, Kageyama (1973) obtained an inequality $b \geqq v + t - 1$. The main purpose of this note is to improve $b \geqq v + t - 1$ to $b \geqq \max \{v + t - 1, (m^2\lambda + m)/\mu^2\}$. This inequality is also improved further for a $\mu$-resolvable BIBD which is not affine $\mu$-resolvable.