On some infinite series of \((r,1)\)-designs
An \((r,1)\)-design is a linear space with \(v\) points where every point is on exactly \(r\) lines. Let \(\Delta = v - r\) be the class of the \((r,1)\)-designs. By examining certain diophantine equations in terms of \(r_i\), the number of \(i\)-lines through a point, the possible ``types'' of points can be determined. Elementary counting arguments then provide necessary conditions on the parameters and point types for the existence of the design. It is shown that for \(\Delta \leq 4\) these necessary conditions are also sufficient. For the case \(\Delta = 5\) some infinite series of \((r,1)\)-designs are constructed. With larger \(\Delta\) values there are examples where the ``obvious'' necessary conditions are not sufficient. The paper is not self-contained and the reader would have to seek for the meaning of some notation and various essential arguments in some of the author's other works.
scientific article; zbMATH DE number 1395001