Some Regular Steiner $2$-Designs with Block Size $4$

Abstract

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 $p\equiv 13\mathrm{mod}24$. We extend the theorem to primes $p\equiv 1\mathrm{mod}24$ 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 $Z_2^2\oplus Z_p$.