Modified linear dependence and the capacity of a cyclic graph

AbstractIn 1956 Shannon raised a problem in information theory, which amounts to this geometric question: How many n-dimensional cubes of width 2 can be packed in the n-dimensional torus described by the nth power of the cyclic group Cm? The present paper examines this question in the special circumstance that the set of centers of the cubes form a subgroup—that is, a lattice packing. In this case, the machinery of vector spaces is available when m is a prime. This approach introduces a modified definition of linear independence, obtains some known results with its aid, and suggests a promising direction for future computation and theory. The paper concludes by showing that, in return, combinatorial information can yield results about finite vector spaces