Communicating over the Torn-Paper Channel
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We consider the problem of communicating over a channel that randomly "tears" the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length n and pieces of length Geometric(p_n), we characterize the capacity as C = e^-α, where α = lim_n→∞ p_n log n. Our results show that the case of Geometric(p_n)-length fragments and the case of deterministic length-(1/p_n) fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.
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