Recovering a Message from an Incomplete Set of Noisy Fragments
We consider the problem of communicating over a channel that breaks the message block into fragments of random lengths, shuffles them out of order, and deletes a random fraction of the fragments. Such a channel is motivated by applications in molecular data storage and forensics, and we refer to it...
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Zusammenfassung: | We consider the problem of communicating over a channel that breaks the
message block into fragments of random lengths, shuffles them out of order, and
deletes a random fraction of the fragments. Such a channel is motivated by
applications in molecular data storage and forensics, and we refer to it as the
torn-paper channel. We characterize the capacity of this channel under
arbitrary fragment length distributions and deletion probabilities. Precisely,
we show that the capacity is given by a closed-form expression that can be
interpreted as F - A, where F is the coverage fraction ,i.e., the fraction of
the input codeword that is covered by output fragments, and A is an alignment
cost incurred due to the lack of ordering in the output fragments. We then
consider a noisy version of the problem, where the fragments are corrupted by
binary symmetric noise. We derive upper and lower bounds to the capacity, both
of which can be seen as F - A expressions. These bounds match for specific
choices of fragment length distributions, and they are approximately tight in
cases where there are not too many short fragments. |
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DOI: | 10.48550/arxiv.2407.05544 |