This paper considers a multigateway multibeam satellite system with multiple feeds per beam. In these systems, each gateway serves a set of beams (cluster) so that the overall data traffic is generated at different geographical areas. Full frequency reuse among beams is considered so that interference mitigation techniques are mandatory. Precisely, this paper aims at designing the precoding scheme which, in contrast to single gateway schemes, entails two main challenges. First, the precoding matrix shall be separated into feed groups assigned to each gateway. Second, complete channel state information (CSI) is required at each gateway, leading to a large communication overhead. In order to solve these problems, a design based on a regularized singular value block decomposition of the channel matrix is presented so that both inter-cluster (i.e. beams of different clusters) and intra-cluster (i.e. beams of the same cluster) interference is minimized. In addition, different gateway cooperative schemes are analysed in order to keep the inter-gateway communication low. Furthermore, the impact of the feeder link interference (i.e. interference between different feeder links) is analysed and it is shown both numerically and analytically that the system performance is reduced severally whenever this interference occurs even though precoding reverts this additional interference. Finally, numerical simulations are shown considering the latest fixed broadband communication standard DVB-S2X so that the quantized feedback effect is evaluated. The proposed precoding technique results to achieve a performance close to the single gateway operation even when the cooperation among gateways is low.