Cyclotomic association schemes of broad classes and applications to the construction of combinatorial structures




Cyclotomy, difference sets, partial difference sets, association schemes


In 2010, G. Fernández, R. Kwashira and L. Martı́nez gave a new cyclotomy on any product R=ni=1 Fqi, where Fqi is a finite field with qi elements. They defined a certain subgroup H of the group of units of this product ring R for which the quotient is cyclic. The orbits of the corresponding multiplicative action of the subgroup on the additive group of R are of two types:

  1. The cyclotomic cosets of the quotient of the group of units of R over the subgroup H.

  2. The n-tuples with arbitrary non-zero elements in positions indicated by a proper subset S of {1, …, n} and zeroes elsewhere.

In this paper, we introduce and study a fusion of a class of association schemes derived from the mentioned cyclotomy. The association schemes that we are proposing correspond with a fusion of orbits associated to subsets S of {1, …, n} of the same cardinality. We call standard cyclotomic association schemes of broad classes to these association schemes. The fusion corresponds to the operation of adding to the permutation group that determines the original association scheme the permutations of R induced by the permutations of the symmetric group Sn.

We use these association schemes to obtain sporadic examples and infinite families of difference sets and partial difference sets.





The Marston Conder Issue of ADAM