This paper discusses an ongoing study that uses a recent development in communication control technology to implement hybrid hypercube structures. These architectures are similar to binary hypercubes, but they also provide added connectivity between the processors. This added connectivity increases communication reliability while decreasing the latency of interprocessor message passing. Because these factors directly determine the speed that can be obtained by multiprocessor systems, these architectures are attractive for applications such as remote exploration and experimentation, where high performance and ultrareliability are required. This paper describes and enumerates these architectures and discusses how they can be implemented with a modified version of the hyperswitch communication network (HCN). The HCN is analyzed because it has three attractive features that enable these architectures to be effective: speed, fault tolerance, and the ability to pass multiple messages simultaneously through the same hyperswitch controller.