A theorem on the entropy of certain binary sequences and applications--I

In this, the first part of a two-part paper, we establish a theorem concerning the entropy of a certain sequence of binary random variables. In the sequel we will apply this result to the solution of three problems in multi-user communication, two of which have been open for some time. Specifically we show the following. Let X and Y be binary random n -vectors, which are the input and output, respectively, of a binary symmetric channel with "crossover" probability p₀. Let H\X\ and H\ Y\ be the entropies of X and Y, respectively. Then equation split 1n H\X\ h (₀), 0 ₀ & 1, \\ \& 1nH\Y\ h (₀ (1 - p₀) + (1 - ₀) p₀) split equation where h () = - - (1 -) (l -), 0 1.