Cyclic designs with block size 4 and related optimal optical orthogonal codes
In this paper an explicit construction is given for a cyclic \((4p,1,1)\)-BIBD if \(p\) is a prime such that either \(p\equiv 1\pmod{12}\) or \(p\equiv 7\pmod{12}\), in the latter case with the extra restriction that \(p> (2q^3- 3q^2+1)^2+ 3q^2\), where \(q\) is the smallest prime factor of \((p-1)/6\). It follows that there is a cyclic \((4,1)\)-GDD of type \(4^p\) in these cases. Furthermore, the author constructs a cyclic \((4,1)\)-GDD of type \(6^p\) for any prime \(p> 5\) and a cyclic \((4,1)\)-GDD of type \(8^p\) for any prime \(p\equiv 1\pmod 6\). It is conjectured that there is a cyclic \((4p,4,1)\)-BIBD for any prime \(p\equiv 7\pmod{12}\) such that \(p> 7\).
scientific article; zbMATH DE number 1785704