Some existence theorems for \(t\)-designs
Some conditions on the parameters of a non-trivial \(t\)-design are investigated that give some guarantee of the existence of another \(t\)- design whose block intersection numbers meet certain criteria. Applying these results, the author has discovered new infinite families of \(5\)-\((v,k,\lambda)\) designs in which any two distinct blocks intersect in less than \(k-1\) points, e.g., for \(n \geq 6\) there exists a \(5\)-\((2^n+4,2^{n-1} + 2,(2^n-1) (2^n-2) (2^{n-1}-2))\) design with \(| B \cap B' | < 2^{n-1}+1\) for any two distinct blocks \(B\) and \(B'\). In fact, these are the first examples of infinite families of 5-designs having this property. So far, the known 5-designs with this property are the known 5-Steiner systems whose parameters are \(5\)-\((12,6,1)\), \(5\)-\((24,8,1)\), \(5\)-\((24,6,1)\), \(5\)-\((28,7,1)\), \(5\)-\((48,6,1)\), \(5\)-\((84,6,1)\) and \(5\)-\((72,6,1)\).
scientific article; zbMATH DE number 562603