Dense symmetric networks from linear groups

An algebraic approach to the problem of constructing large networks of bounded degree and diameter is described. Evidence is provided showing that the table of largest known constructions for small values of the two parameters can be improved almost everywhere by methods based on finite groups. In many entries, the constructions are dramatically larger than the best previously known and many of these improvements are in the range of the numbers of processors currently being considered for large parallel processing systems. These constructions, all highly symmetric, can be viewed as belonging to a family of constructions based on vector spaces and their automorphism groups that includes hypercubes and cube-connected cycles as special cases.