On balanced complementation for regular t-wise balanced designs

AbstractVanstone has shown a procedure, called r-complementation, to construct a regular pairwise balanced design from an existing regular pairwise balanced design. In this paper, we give a generalization of r-complementation, called balanced complementation. Necessary and sufficient conditions for balanced complementation which gives a regular t-wise balanced design from an existing regular t-wise balanced design are shown. We characterize those aspects of designs which permit balanced complementation. Results obtained here will be applied to construct regular t-wise balanced designs which are useful in Statistics

The cone condition and t-designs

AbstractThe existence of a t-(v,k,λ) design implies that certain ‘almost constant’ vectors belong to the convex cone generated by the columns of the incidence matrix of t-subsets versus k-subsets of a v-set. We prove that some vectors are not in, or in a few cases are in, this cone—whether a design exists or not. When certain vectors are shown not to be in this cone, the implication is an inequality on the parameters or a condition on the structure of a t-design. We unify a number of known inequalities for t-designs, and derive some new ones concerning t-wise balanced designs, with this approach