Digital transmission with coherent four-dimensional modulation

This paper examines codes for four-dimensional (4-D) modulation and their performance for digital transmission. The signals are defined by
M
points inside a sphere in four-dimensional Euclidian space. Three classes of 4-D codes are presented, and an algorithm is given which yields good 4-D codes of any length. Bounds on symbol error probability are plotted versus symbol-energy-to-noise-density ratio. The performance is shown to exceed that of amplitude- and-phase modulation in two independent two-dimensional channels.