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

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