Wildcard Dimensions, Coding Theory and Fault-Tolerant Meshes and Hybercubes

Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First the authors prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. They then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes they construct d-dimensional meshes and tori with wildcard dimensions. Finally, they show how these constructions can be used to tolerate edge and node faults in mesh and torus networks.