We give a constructive and very simple proof of a theorem by P. L. Check and C. J. Colbourn [Discrete Math. 133 (1994), no. 1-3, 285--289] stating the existence of a cyclic $(4p,4,1)$-BIBD (i.e. regular over $Z_{4p}$) for any prime $p\equiv13\pmod{24}$. We extend the theorem to primes $p\equiv1\pmod{24}$, 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 $Z_2^2\oplus Z_p$.

### Some regular Steiner 2-designs with block size 4

#### Abstract

We give a constructive and very simple proof of a theorem by P. L. Check and C. J. Colbourn [Discrete Math. 133 (1994), no. 1-3, 285--289] stating the existence of a cyclic $(4p,4,1)$-BIBD (i.e. regular over $Z_{4p}$) for any prime $p\equiv13\pmod{24}$. We extend the theorem to primes $p\equiv1\pmod{24}$, 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 $Z_2^2\oplus Z_p$.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11391/22773
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