On few-class $Q$-polynomial association schemes: feasible parameters and nonexistence results

TL;DR

This paper provides tables of feasible parameters for certain classes of $Q$-polynomial association schemes and establishes nonexistence results for some cases by analyzing intersection numbers, advancing understanding of their structural possibilities.

Contribution

It offers the first comprehensive tables of feasible parameters for primitive 3-class and 4- and 5-class $Q$-bipartite association schemes, along with new nonexistence proofs.

Findings

Feasible parameter tables for primitive 3-class schemes.

Feasible parameter tables for 4- and 5-class $Q$-bipartite schemes.

Nonexistence results for certain open cases.

Abstract

We present the tables of feasible parameters of primitive $3$-class $Q$-polynomial association schemes and $4$- and $5$-class $Q$-bipartite association schemes (on up to $2800$, $10000$, and $50000$ vertices, respectively), accompanied by a number of nonexistence results for such schemes obtained by analysing triple intersection numbers of putative open cases.