The cone condition and -designs

The 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.