  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 $s=4$.
