Classification of tight $2s$-designs with $s\ge 2$

Abstract

Tight $2s$-designs are the $2s$-$(v,k,\lambda )$ designs whose sizes achieve the Fisher type lower bound $\left(\genfrac{}{}{0ex}{}{v}{s}\right)$. Symmetric $2$-designs, the Witt $4$-$(23,7,1)$ design and the Witt $4$-$(23,16,52)$ design are tight designs. It has been widely conjectured since 1970s that there are no other nontrivial tight designs. In this paper, we give a proof of this conjecture. In the proof, an upper bound $v\ll s$ is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound $v\gg s(\ln s{)}^2$ is shown by using estimates on prime gaps.