Distance-regular digraphs of girth 4 over an extension ring of Z/4Z

Enomoto-Mena[1] showed that two one-parameter families of distance-regular digraphs of girth 4 could possibly exist. Subsequently Liebler-Mena[2] found an infinite family of such digraphs generated over an extension ring ofZ/4Z. We prove that there are no other solutions except for multiplication by principal units to generate distance-regular digraphs of girth 4 under their method. In order to prove this, we introduce Gauss sums and three kinds of Jacobi sums over an extension ring ofZ/4Z. We give necessary and sufficient conditions for the existence of these digraphs under that method. It turns out that the Liebler-Mena solutions are the only solutions which satisfy the necessary and sufficient conditions. This fact has been conjectured for a time, but has never been proved.