Geodesic
Nonsymmetric primitive translation schemes of prime power order
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 1-20.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

It is well-known that translation schemes of prime order are exactly the cyclotomic schemes. In this paper, we show that there do not exist any two-, three-, or four-class nonsymmetric primitive translation schemes of prime square order. On the other hand, we find new nonsymmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class association schemes. Moreover, we provide an affirmative answer to the following question raised by S. Y. Song [J. Algebr. Comb. 5, No. 1, 47--55 (1996; Zbl 0843.05103)]: are there any other two-class primitive association schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for $q \equiv 5$ (mod 8)? To be more specific, we show that a certain two-class primitive association scheme in the finite field $\mathbb{F}_{37^3}$ constructed by T. Feng and Q. Xiang [J. Comb. Theory, Ser. A 119, No. 1, 245--256 (2012; Zbl 1232.05031)] admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.
Classification : 05E30
Keywords: nonsymmetric primitive translation scheme, cyclotomic scheme, Gauss sum 
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Feng, Tao; Momihara, Koji. Nonsymmetric primitive translation schemes of prime power order. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 1-20. http://geodesic.mathdoc.fr/item/JAC_2015__41_1_a10/