The cone condition and tt-designs

The existence of a t-(v,k,λ)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 tt-subsets versus kk-subsets of a vv-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 tt-design. We unify a number of known inequalities for tt-designs, and derive some new ones concerning tt-wise balanced designs, with this approach.