More Results on the Existence of Simple BIBDs with Number of Elements a Prime Power

Abstract

We study the existence of simple $(q, k, \lambda)$ BIBDs when the number of elements is a prime power $q$ and $\{c_1, c_2\} \cap \{1, 2\}$ is not empty, where $c_1 = \gcd(k, q-1)$ and $c_2 = \gcd(k-1, q-1)$. We show that in many situations the necessary conditions $\lambda (q-1) \equiv 0 \mod (k-1)$, $\lambda q(q-1) \equiv 0 \mod k(k-1)$, and $\lambda \leq \binom{q-2}{k-2}$ are also sufficient for the existence of a simple $(q, k, \lambda)$ BIBD. These new results improve the valid range of simple BIBDs.