Capacity of the Torn Paper Channel with Lost Pieces

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
$n$
, discarding fragments shorter than
$\log(n)$
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”.