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

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2 Author(s)

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 p_0 . Let H{X} and H{ Y} be the entropies of X and Y , respectively. Then begin{equation} begin{split} frac{1}{n} H{X} geq h(alpha_0), qquad 0 leq alpha_0 &leq 1, Rightarrow \ qquad qquad &qquad frac{1}{n}H{Y} geq h(alpha_0(1 - p_0) + (1 - alpha_0)p_0) end{split} end{equation} where h(\lambda ) = -\lambda \log \lambda - (1 - \lambda ) \log (l - \lambda ), 0 \leq \lambda \leq 1 .

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Information Theory, IEEE Transactions on  (Volume:19 ,  Issue: 6 )