Information rate in a continuous channel for regular-simplex codes

M
equally probable symbols may be encoded as continuous waveforms whose sample values are the coordinates of the vertices of a regular simplex in
N
dimensions. These vertices or code points, therefore, are equally spaced, and the resulting waveforms are adapted particularly to low signal-to-noise communication systems. An upper bound for the information rate in an additive Gaussian noise channel, based on the use of regular-simplex coded waveforms, has been calculated. The results indicate that for signal-to-noise energy ratios less than approximately unity and for error probabilities ranging from
10^{-2}
to
l0^{-8}
, this rate is a sizable percentage of the ideal rate which has been derived by Shannon and Tuller.