We consider a space communication network consisting of Geosynchronous Earth Orbit satellites (GEOSs) and Low Earth Orbit satellites (LEOSs). In case of no direct communication link between two GEOSs, the data exchange between them is through relay by the LEOSs. In particular, the source GEOS sends coded data to multiple LEOSs based on the distributed storage framework. The destination GEOS then retrieves certain amount of data from each LEOS for data reconstruction. For the GEOS-LEOS downlink, a regenerating-code-based transmission scheme is optimized to guarantee data reconstructability, where the transmission power allocation to the LEOSs is proposed to minimize the total transmission energy. We also consider the power allocation to minimize the total transmission time given the total transmission energy. For the LEOS-GEOS uplink, a flexible partial-downloading coding transmission scheme is proposed to guarantee data reconstructability, where the joint uploaded-data size and power allocations are proposed to minimize the total transmission energy or the total transmission time. Extensive simulation results are presented to evaluate the proposed algorithms and show that regenerating code can achieve lower transmission energy and shorter transmission time for data regeneration than conventional maximum-distance separable (MDS) code.