In his thesis, Delsarte defines a duality construction for association schemes. Nevertheless, this duality construction works only if some special regularity condition is satisfied. I find the condition hard to grasp. It seems just like a technical assumption in order to make it work. A particular case is the duality of translation association schemes associated to abelian groups (Thm 2.9), which works always and is described also in various other texts such as this or this.

Is the general construction of Delsarte really more general or is it true that actually any association scheme that has this kind of dual must be a translation association scheme?

In his thesis (1973), P. Delsarte defines a duality construction for association schemes. Nevertheless, this duality construction works only if some special regularity condition is satisfied. I find the condition hard to grasp. It seems just like a technical assumption in order to make it work. A particular case is the duality of translation association schemes associated to abelian groups (Theorem 2.9), which works always and is described also in various other texts such as this or this.

Is the general construction of Delsarte really more general or is it true that actually any association scheme that has this kind of dual must be a translation association scheme?