摘要 : We consider the discrete memoryless symmetric primitive relay channel, where, a source X wants to send information to a destination Y with the help of a relay Z and the relay can communicate to the destination via an error-free di... 展开 We consider the discrete memoryless symmetric primitive relay channel, where, a source X wants to send information to a destination Y with the help of a relay Z and the relay can communicate to the destination via an error-free digital link of rate R0, while Y and Z are conditionally independent and identically distributed given X. We develop two new upper bounds on the capacity of this channel that are tighter than existing bounds, including the celebrated cut-set bound. Our approach significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on the blowing-up lemma to analyze the probabilistic geometric relations between the typical sets of the n-letter random variables associated with a reliable code for communicating over this channel. These relations translate to new entropy inequalities between the n-letter random variables involved. As an application of our bounds, we study an open question posed by (Cover, 1987), namely, what is the minimum rate R0* needed for the Z-Y link in order for the capacity of the relay channel to be equal to that of the broadcast cut. We consider the special case when the X-Y and X-Z links are both binary symmetric channels. Our tighter bounds on the capacity of the relay channel immediately translate to tighter lower bounds for R0*. More interestingly, we show that when p → 1/2, R0* ≥ 0.1803; even though the broadcast channel becomes completely noisy as p → 1/2 and its capacity, and therefore the capacity of the relay channel, goes to zero, a strictly positive rate R0 is required for the relay channel capacity to be equal to the broadcast bound. Existing upper bounds on the capacity of the relay channel, and the cut-set bound in particular, would rather imply R0* → 0, while achievability schemes require R0* → 1. We conjecture that R0* → 1 as p → 1/2. 收起