We first explain the historical and logical relations of hyperstructures introduced by M. Krasner and R. Rota, and generalized by T. Vougiouklis. Then, with our new algorithm based on our previous results on hypergroups and H_v-groups of order 2, 3 and 4, we enumerate hyperrings and H_v-rings. More precisely, we found 63 hyperrings of order 2, 875 H_v-rings of order 2 and 33,277,642 hyperrings of order 3. Finally, in this new context, we study a new connection between groups and hypergroups via the notion of duality.