From a normative viewpoint, there is no compelling reason for preferring the weak over the strong core, and vice versa. However, the situation changes significantly from a mechanism design perspective. We work in a rights structures environment, where the role of the social planner is to allocate rights to individuals or coalitions which allow them to change the status-quo state. While coalitions are irrelevant for implementation in weak core (Koray and Yildiz in J. Econ. Theory 176:479-502, 2018; Korpela et al. in J. Econ. Theory 185:104953, 2020), our results show that they are fundamental for implementation in strong core. We fully characterize the implementation of social choice rules in strong core to outline this distinction. For robustness, we also characterize double implementation in weak and strong core which we show to be equivalent to implementation in weak core. Finally, we show that this equivalence breaks down in the more realistic case of implementation by codes of rights, where the set of states coincides with the set of outcomes.