Group Codes for Gaussian Channel

In a communication model first introduced by Kotel'nikov 1 in 1947, and independently by Shannon 2 in 1948, and since studied by many authors, 3 - 2 2 messages for transmission are represented by vectors in a Euclidean space, S,, , of n dimensions called signal space. In this model, known as the Gaussian channel, when X is transmitted, the received signal is represented by a vector Z = X + Y which consists of the sum of the sent vector and a noise vector Y whose components are independent Gaussian variates with mean zero and variance a 2 . Some physical circumstances t h a t lead to this model, as well as further details, can be found in Refs. 3, 10, and 13. An equal-energy block code of size M for use on this Gaussian channel is a collection of M distinct vectors Xa , X2 , · · · , X^ in signal space all of the same length. We shall always suppose M ^ n and that the vectors span S,, . The length of the vectors serves to define an important parameter S called the average power of the code through the equation = |X,| 2 . (1)