Quasi-Symmetric 3-Designs and Elliptic Curves
A.R. Calderbank, Patrick Morton · SIAM Journal on Discrete Mathematics · 1990
A quasi-symmetric t-design is a t-design with two block intersection sizes p and q (where $p < q$). Quasi-symmetric 3-designs are classified with $p = 1$. The only nontrivial examples are the 4-(23, 7, 1) Witt design, and its residual, a 3-(22, 7, 4) design. This proves a conjecture of Sane and Shrikhande. The method is to reduce the classification problem to that of finding all integer points on the elliptic curves $y^2 = x^3 - 11x^2 + 32x$ and $y^2 = x^3 - 4x + 4$.