A prospect for a general method of constructing \(t\)-designs
By a \(t\)-design we mean a collection of sets \(S=S(\lambda;t,k,v)\) so that each set in \(S\) is a \(k\)-element subset of a fixed \(v\)-element set \(X\) and so that each \(t\)-element subset of \(X\) is a subset of exactly \(\lambda\) of the sets in \(S\). If \(S\) is a design then the numbers \[ \lambda_ i={{v-i\choose t-i}\over{k-i\choose t-i}} \] must be integers for \(0\leq i\leq t\). Thus, the requirement that these numbers are integers is a necessary condition for the existence of a \(t\)-design with parameters \((\lambda;t,k,v)\). This requirement is not sufficient; but it is sufficient for the existence of a signed \(t\)-design, i.e. one in which the sets in \(S\) are assigned integers multiplicities and the condition on \(t\)-element subsets is altered to read: the multiplicities of the sets in \(S\) which contain a \(t\)-set must sum to \(\lambda\). Signed \(t\)-designs were invented to aid in the construction of (unsigned) \(t\)-designs. The authors review the theoretical knowledge of \(t\)-designs and signed \(t\)-designs and produce two algorithms: one for constructing signed \(t\)- designs and one for altering these signed \(t\)-designs to get (unsigned) \(t\)-designs with the same set of parameters.
scientific article; zbMATH DE number 167438
scientific article