Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

Abstract

In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in $3$ variables. We demonstrate it by giving a classification of finite subsets that are spherical $2$-distance sets and spherical $\{4,2,1\}$-designs at the same time.