Nonexistence results for tight block designs

Recall that combinatorial 2s-designs admit a classical lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b \ge\binom{v}{s}$\end{document} 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≥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 larger s. Additionally, we obtain strong necessary conditions for existence in the difficult case s=4.