Quasimultiples of geometric designs
A block design \(D= (P,B)\) is a collection \(B\) of \(k\)-subsets (called blocks) of a \(v\)-set \(P\) such that every \(t\)-subset of \(P\) is contained in exactly \(\lambda\) blocks, for some positive constant \(\lambda\). Such a design is often referred to as a \(t\)-\((v, k,\lambda)\) design. With this notation a \(t\)-\((v, k,s\lambda)\) design is called an \(s\)-fold quasimultiple of a \(t\)-\((v,k,\lambda)\) design, for any integer \(s\geq 2\). Quasimultiple projective and affine planes (that is \(2\)-\((n^2+ n+ 1,n +1,\lambda)\) and \(2\)-\((n^2,n,\lambda)\) designs, respectively) have received considerable attention in recent years and it is conjectured that quasidouble affine planes exist for all values of \(n\). In this paper, quasimultiples of affine planes, inversive planes, and unitals from geometrical configurations in various finite geometries are constructed. All designs have \(k= \lambda.\) These designs are simple (no repeated blocks), and all appear to be irreducible in the sense that a quasimultiple of type \(P\) will have no subdesigns isomorphic to \(P\). This is verified for small orders by computer computations.
scientific article; zbMATH DE number 2093773
scientific article