Capacity of the Torn Paper Channel with Lost Pieces

Aditya Narayan Ravi, Alireza Vahid, Ilan Shomorony

Abstract: We study the problem of transmitting a message over a channel that randomly breaks the message block into small fragments, deletes a subset of them, and shuffles the remaining fragments. We characterize the capacity of the binary torn-paper channel under arbitrary fragment length distribution and fragment deletion probabilities. We show that, for a message with block length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> , discarding fragments shorter than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\log(n)$</tex> does not affect the achievable rates, and that the capacity is given by a simple closed-form expression that can be understood as “coverage minus reordering-cost”.