Probability of Error for Optimal Codes in a Gaussian Channel

Consider a communication channel of the following type: Once each second a real number may be chosen at the transmitting point. This number is transmitted to the receiving point but is perturbed by an additive gaussian noise, so that the fth real number, s , , is received as Si + Xi. The Xi are assumed independent gaussian random variables all with the same variance N. A code word of length n for such a channel is a sequence of n real numbers ( s i , s2, · · · , s,,). This may be thought of geometrically as a point in n-dimensional Euclidean space. The effect of noise is then to move this point to a nearby point according to a spherical gaussian distribution. A block code of length n with M words is a mapping of the integers 1, 2, · · · , M into a set of M code words Wi, w2, · · · , wM (not necessarily 611 012 T H E B E L L SYSTEM T E C H N I C A L JOU11NAL, MAY l(Jo!) all distinct). Thus, geometrically, a block code consists of a collection of M (or less) points with associated integers. It may be thought of as a way of transmitting an integer from 1 to M to the receiving point (by sending the corresponding code word). A decoding system for such a code is a partitioning of the w-dimensional space into M subsets corresponding to the integers from 1 to M. This is a way of deciding, at the receiving point, on the transmitted integer. If the received signal is in subset S i , the transmitted message is taken to be integer i. We shall assume throughout that all integers from 1 to M occur as messages with equal probability 1/il/.