Rings associated to finite projective planes and thier isomorphisms

In this paper we announce an explicit form of the standard basis of the 2-extended ring associated to the cellular ring generated by the incidence graph of a finite projective plane.This enables us to find the first example of a distance-regular graph satisfying the 6-condition which is not a distance-transivite one. One more corollary of the result obtained is that the cellular rings of any two projective planes of the same order are 2-isomorphic. This implies that if there exist at least two nonisomorphic and nondual to each other projective planes of a given order, then the separability number of any projective plane of this order is greater or equal to 3 and, moreover, it is equal to 3 for a Galois plane.