In his [thesis][1] (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][2] or [this][3].

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?


  [1]: https://users.wpi.edu/~martin/RESEARCH/philips.pdf
  [2]: https://www.math.uwaterloo.ca/~cgodsil/pdfs/assoc2.pdf
  [3]: https://www.sciencedirect.com/science/article/pii/S0195669808002448