August 17, 2025

A new example of non-amorphous association schemes

Key Points

Every nontrivial relation in the new example forms a strongly regular graph, enhancing understanding of graph theory.
The example is a fusion scheme of the cyclotomic scheme of class 75 on GF(2^20).
An infinite family of parameters for association schemes expands existing knowledge and applications.
These findings suggest a deeper relationship between non-amorphous schemes and strongly regular graphs in combinatorial design.

Abstract

E. R. van Dam gave an example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class 45 on ¥GF (2¹2). The aim of this paper is to present a new example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class 75 on ¥GF (2²0). We also propose an infinite family of parameters of association schemes containing both of these two examples.

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