Nonexistence Results for Tight Block Designs

TL;DR

This paper proves the nonexistence of tight 2s-designs for s between 5 and 9 using polynomial bounds and computer search, and provides necessary conditions for existence when s=4.

Contribution

It applies the Smith Bound to quantify nonexistence and conducts a computer search to confirm no tight 2s-designs exist for 5 ≤ s ≤ 9, extending understanding of design nonexistence.

Findings

No tight 2s-designs for 5 ≤ s ≤ 9

Quantitative bounds on polynomial zeros used for nonexistence proof

Necessary conditions established for s=4

Abstract

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 $s=4$.