Some Regular Steiner \(2\)-Designs with Block Size \(4\)

We give a constructive and very simple proof of a theorem by Chech and Colbourn [7] stating the existence of a cyclic \((4p, 4, 1)\)-BIBD (i.e. regular over \({Z}_{4p}\)) for any prime \(p \equiv 13 \mod 24\). We extend the theorem to primes \(p \equiv 1 \mod 24\) although in this case the construction is not explicit. Anyway, for all these primes \(p\), we explicitly construct a regular \((4p, 4, 1)\)-BIBD over \({Z}_{2}^{2} \oplus {Z}_p\).