A categorical proof of the nonexistence of (120, 35, 10)-difference sets

Abstract

A difference set with parameters $(v,k,\lambda )$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $\lambda$ of occurrences of $g\in G$ as the ratio $d^{-1}{d}^{\prime }$ in distinct pairs $(d,d^{\prime })\in D\times D$ is independent of $g$. We prove the nonexistence of $(120,35,10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets. Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints.

16 pages, no figures