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

TL;DR

This paper proves the nonexistence of a specific difference set with parameters (120, 35, 10), resolving a 70-year-old open problem using advanced algebraic and combinatorial methods.

Contribution

It introduces a novel approach combining association schemes, equi-distributed functions, and linear programming to solve a longstanding combinatorial problem.

Findings

Proves nonexistence of (120, 35, 10)-difference sets.

Develops a generalized framework for difference sets using association schemes.

Applies linear programming with quadratic constraints to combinatorial existence problems.

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 $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ 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.