顶点素数幂的非对称本原平移格式
Nonsymmetric primitive translation schemes on prime power number of vertices
摘要：
众所周知，顶点数为素数的平移格式就是割圆格式。在这篇文章中，我们证明了在最多四个类的素数正方形顶点上不存在非对称的本原平移方案。另一方面，我们发现新的非对称的四类和五类结合方案的割圆作为裂变计划的某些对称的三类计划。此外，我们还肯定地回答了Song \cite{song_2}提出的以下问题：除了$q \equiv5 \pmod{8}$的指数为2的分圆方案外，是否还有其他的两类本原方案允许对称化裂变方案？更具体地说，我们证明了Feng和Xiang在\cite{fx}中构造的有限域$\F_{37^3}$中的一类两类本原概型允许四类裂变概型.这个分裂方案被实现为指数为28的分圆方案的融合方案。
英文摘要：
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.
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