This paper examines codes for four-dimensional (4-D) modulation and their performance for digital transmission. The signals are defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> 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.
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Welti et al. (1974) studied this question.
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