Torn-Paper Coding

• Published in: IEEE transactions on information theory Vol. 67; no. 12; pp. 7904 - 7913
• Main Authors: Shomorony, Ilan, Vahid, Alireza
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Codes; Communication; data storage; Decoding; DNA; DNA storage; Encoding; Fragments; Indexes; Sequential analysis; Torn paper; Transmitters; unordered communication
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3120920

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 <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and pieces of length <inline-formula> <tex-math notation="LaTeX">{\mathrm{ Geometric}}(p_{n}) </tex-math></inline-formula>, we characterize the capacity as <inline-formula> <tex-math notation="LaTeX">C = e^{-\alpha } </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">\alpha = \lim _{n\to \infty } p_{n} \log n </tex-math></inline-formula>. Our results show that the case of <inline-formula> <tex-math notation="LaTeX">{\mathrm{ Geometric}}(p_{n}) </tex-math></inline-formula>-length fragments and the case of deterministic length-<inline-formula> <tex-math notation="LaTeX">(1/p_{n}) </tex-math></inline-formula> 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.