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Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$of the concept of congruence in $Z$. Here we use this approach to present afamily of such digraphs, which, for every fixed value of the degree, haveasymptotically large number of vertices as the diameter increases. Up to now,the best known asymptotically dense results were all non-constructive.

Abelian Cayley digraphs with asymptotically large order for any given degree