Geometric properties of a large set of random signals

For a bandwidth-limited, power-limited communication channel corrupted by additive white Gaussian noise, Shannon (1949) showed that, at any information rate below the channel capacity, an arbitrarily low error probability can be obtained when the transmitted signal is selected from a set a of M independent, white-Gaussian-noise-like waveforms having the prescribed power and bandwidth. He utilized a space whose points represent transmitted or received signals. Since for each possible received signal there is a most likely transmitted signal, this space is divided into decision regions {R_t}, one associated with each member s_t (i=1, 2,..., M) of set s_t This article investigates some of the geometrical properties of these decision regions: their shape, size and number of faces, distances from s_t to R_t´s nearest face or edges of various sorts, etc., thus providing insight into how random coding succeeds in achieving arbitrarily good performance