We introduce and study a family of interconnection schemes, the Midimew networks, based on circulant graphs of degree 4. A family of such circulants is determined and shown to be optimal with respect to two distance parameters simultaneously, namely maximum distance and average distance, among all circulants of degree 4.. These graphs are regular, point-symmetric, and maximally connected, and one such optimal graph exists for any given number of nodes. The proposed interconnection schemes consist of mesh-connected networks with wrap-around links, and are isomorphic to the optimal distance circulants previously considered. Ways to construct one such network for any number of nodes are shown, their good properties to build interconnection schemes for multicomputers are examined, and some interesting particular cases are discussed. The problem of routing is also addressed, and a basic algorithm is provided which is adequate for implementing the routing policy required to convey messages, traversing shortest paths between nodes.