Nonsymmetric primitive translation schemes on prime power number of vertices

TL;DR

This paper investigates nonsymmetric primitive translation schemes on prime power vertices, showing their nonexistence on prime square vertices with few classes, and constructs new association schemes from cyclotomy, answering a question about fission schemes.

Contribution

It proves the nonexistence of certain nonsymmetric schemes and constructs new association schemes from cyclotomy, expanding understanding of primitive translation schemes.

Findings

No nonsymmetric primitive translation schemes on prime square vertices with up to four classes.

New non-symmetric four- and five-class schemes derived from cyclotomy.

A two-class primitive scheme admits a four-class fission scheme as a fusion of a cyclotomic scheme.

Abstract

It is well-known that translation schemes on prime number of vertices are exactly the cyclotomic schemes. In this current paper, we show that there are no nonsymmetric primitive translation schemes on prime square vertices with at most four classes. On the other hand, we find new non-symmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class schemes. Moreover, we provide an affirmative answer to the following question raised by Song \cite{song_2}: Are there any other two-class primitive schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for $q \equiv5 \pmod{8}$? To be more specific, we show that a certain two-class primitive scheme in the finite field $\F_{37^3}$ constructed by Feng and Xiang in \cite{fx} admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.