A forbidden rate region for generalized cross constellations

An analysis of the generalized cross constellation (GCC) is presented and a new perspective on its coding algorithm is described. We show how the GCC can be used to address generic sets of symbol points in any multidimensional space through an example based on the matched spectral null coding used in magnetic recording devices. We also prove that there is a forbidden rate region of fractional coding rates that are practically unrealizable using the GCC construction. We introduce the idea of a constellation tree and show how its decomposition can be used to design GCCs matching desired parameters. Following this analysis, an algorithm for designing the optimal rate GCC from a restriction on the maximum size of its constellation signal set is given, and a formula for determining the size of the GCC achieving a desired coding rate is derived. We finish with an upper bound on the size of the constellation expansion ratio.