Some New Direct Constructions for $(\nu ,\{5,\omega \ast \},1)$ PBDs

Abstract

For $\omega \le 33$, the known necessary conditions for existence of a $(\nu ,\{5,{\omega }^{\ast }\},1)$ PBD, namely $\nu ,\omega \equiv 1\mathrm{mod}4$, $\nu \ge 4\omega +1$ and $\nu \equiv \omega$ or $4\omega +1\mathrm{mod}20$ are known to be sufficient in all but 26 cases. This paper provides several direct constructions which reduce the number of exceptions to 8.