A survey of results for deletion channels and related synchronization channels

The purpose of this survey is to describe recent progress in the study of the binary deletion channel and related channels with synchronization errors, including a clear description of open problems in this area, with the hope of spurring further research. As an example, while the capacity of the binary symmetric error channel and the binary erasure channel have been known since Shannon, we still do not have a closed-form description of the capacity of the binary deletion channel. We highlight a recent result that shows that the capacity is at least $(1-p)/9$ when each bit is deleted independently with fixed probability $p$.

[1]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[2]  Jeffrey D. Ullman,et al.  On the capabilities of codes to correct synchronization errors , 1967, IEEE Trans. Inf. Theory.

[3]  Masakazu Jimbo,et al.  An Iteration Method for Calculating the Relative Capacity , 1979, Inf. Control..

[4]  Khaled Abdel-Ghaffar Capacity per unit cost of a discrete memoryless channel , 1993 .

[5]  David Zuckerman,et al.  Asymptotically good codes correcting insertions, deletions, and transpositions , 1997, SODA '97.

[6]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[7]  Michael Mitzenmacher,et al.  A Note on Low Density Parity Check Codes for Erasures and Errors , 1998 .

[8]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[9]  Suhas Diggavi,et al.  On transmission over deletion channels , 2001 .

[10]  David J. C. MacKay,et al.  Reliable communication over channels with insertions, deletions, and substitutions , 2001, IEEE Trans. Inf. Theory.

[11]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[12]  M. Mitzenmacher,et al.  Concatenated codes for deletion channels , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[13]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[14]  Sampath Kannan,et al.  Reconstructing strings from random traces , 2004, SODA '04.

[15]  Sampath Kannan,et al.  More on reconstructing strings from random traces: insertions and deletions , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[16]  Michael Mitzenmacher,et al.  Polynomial Time Low-Density Parity-Check Codes With Rates Very Close to the Capacity of the $q$-ary Random Deletion Channel for Large $q$ , 2006, IEEE Transactions on Information Theory.

[17]  Lara Dolecek,et al.  A Synchronization Technique for Array-based LDPC Codes in Channels With Varying Sampling Rate , 2006, 2006 IEEE International Symposium on Information Theory.

[18]  Michael Mitzenmacher,et al.  On Lower Bounds for the Capacity of Deletion Channels , 2006, IEEE Transactions on Information Theory.

[19]  Michael Mitzenmacher,et al.  A Simple Lower Bound for the Capacity of the Deletion Channel , 2006, IEEE Transactions on Information Theory.

[20]  Michael Mitzenmacher,et al.  Improved Lower Bounds for the Capacity of i.i.d. Deletion and Duplication Channels , 2007, IEEE Transactions on Information Theory.

[21]  John J. Metzner Packet-Symbol Decoding for Reliable Multipath Reception with No Sequence Numbers , 2007, 2007 IEEE International Conference on Communications.

[22]  Rina Panigrahy,et al.  Trace reconstruction with constant deletion probability and related results , 2008, SODA '08.

[23]  Krishnamurthy Viswanathan,et al.  Improved string reconstruction over insertion-deletion channels , 2008, SODA '08.

[24]  Michael Mitzenmacher,et al.  Capacity Bounds for Sticky Channels , 2008, IEEE Transactions on Information Theory.

[25]  Dario Fertonani,et al.  Novel bounds on the capacity of binary channels with deletions and substitutions , 2009, 2009 IEEE International Symposium on Information Theory.

[26]  Eleni Drinea,et al.  Directly Lower Bounding the Information Capacity for Channels With I.I.D. Deletions and Duplications , 2010, IEEE Transactions on Information Theory.

[27]  Zhenming Liu,et al.  Codes for Deletion and Insertion Channels With Segmented Errors , 2010, IEEE Transactions on Information Theory.