Nonexistence results for tight block designs
Recall that combinatorial 2s-designs admit a classical lower bound \(b \ge\binom{v}{s}\) on their number of blocks, and that a design meeting this bound is called tight. Along-standing result of Bannai is that there exist only finitely many nontrivial tight 2s-designs for each fixed s?5, although no concrete understanding of finitely many is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for 5?s?9, although the same analysis could in principle be extended to largers. Additionally, we obtain strong necessary conditions for existence in the difficult case s=4.
tight design symmetric design orthogonal polynomials delsartetheory