2023/12/22 by Xiang, Ziqing
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.14778
Tight 2 s-designs are the 2 s-(v, k, λ) designs whose sizes achieve the Fisher type lower bound v \choose s. 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 ≪ s is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound v ≫ s (ln s)2 is shown by using estimates on prime gaps.