Lee distance,Gray codes,and the torus

The torus is a topology that is the basis for the communication network of several multicomputers in use today. This paper briefly explores several topological characteristics of a generalized torus network using concepts from Coding theory and Graph theory. From Coding theory, the Lee distance metric and Gray codes are extended to mixed radix numbers. Lee distance is used to state the number and length of disjoint paths between two nodes in a torus. In addition, a function mapping a sequence of mixed radix numbers to a mixed radix Gray code sequence is described; and, provided at least one radix is even, this sequence is used to embed in the torus a cycle of any even length, including a Hamiltonian cycle. The torus is defined both as a cross product of cycles and using Lee distance. The graph-theoretic definition of a torus leads to a simple single node broadcasting algorithm, which is described in the last section. This revised version was published online in June 2006 with corrections to the Cover Date.