On the Optimality of the Regular Simplex Code

Since its introduction by Shannon1 and Kotel'nikov2 nearly 20 years ago, the geometric representation of signals has played an important role in communication theory.* By this scheme, a variety of physically different time-continuous communication systems can all be reduced to the same geometric model. The problem of finding optimal signals for transmission then becomes a geometric one. This paper solves one such problem. In the model in question, signals to be transmitted are represented as points, or vectors from the origin, in a suitable finite dimensional Euclidean signal space £,, . The energy of any signal in S,, is proportional to the length of its representative vector; the bandwidth of the communication system is proportional to the dimension n of the signal space. Received signals are also represented by vectors in 8,, and the difference Z = Y -- X between a transmitted signal X and the corresponding received signal Y is a vector random variable representative of the noise encountered during transmission. In a model commonly considered, the probability density of Z depends only on its magnitude, i.e., · · · , * ) = / ( | Z |), (1)