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. A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, 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 $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$. Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$.