Nonasymptotic upper bounds for deletion correcting codes

IEEE Transactions on Information Theory

Volumn 59, Issue 8, 2013, Pages 5115-5130

  Kulkarni, Ankur A ^a,b   Kiyavash, Negar ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b INDIAN INSTITUTE OF TECHNOLOGY BOMBAY   (India)

Author keywords

Deletion channel; hypergraphs; integer linear programming; linear programming relaxation; multiple deletion correcting codes; nonasymptotic bounds; single deletion correcting codes; Varshamov Tenengolts codes

Indexed keywords

CORRECTING CODES; DELETION CHANNELS; HYPER GRAPH; INTEGER LINEAR PROGRAMMING; LINEAR PROGRAMMING RELAXATION; NON-ASYMPTOTIC BOUNDS; VARSHAMOV-TENENGOLTS CODES;

LINEAR PROGRAMMING;

INTEGER PROGRAMMING;

EID: 84880519203     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2013.2257917     Document Type: Article

References (45)

1. N. J. A. Sloane, "On single-deletion-correcting codes," in Codes Designs: Proc. Conf. Honor. Professor Dijen K. Ray-Chaudhuri on the Occasion of His 65th Birthday, The Ohio State University, May 18-21, 2000, Walter de Gruyter, 2002.
2. V. I. Levenshtein, "Binary codes capable of correcting deletions, insertions, and reversals," Soviet Phys. Doklady, vol. 10, no. 8, pp. 707-710, 1966.
3. R. R. Varshamov and G. M. Tenengolts, "Codes which correct single asymmetric errors (in Russian)," Avtomatika i Telemekhanika, vol. 6, no. 2, pp. 286-290, 1965.
4. N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs Jul. 2011 [Online]. Available: http://neilsloane.com/doc/graphs.html
5. G. M. Tenengolts, "Nonbinary codes, correcting single deletion or insertion," IEEE Trans. Inf. Theory, vol. 30, no. 5, pp. 766-769, Sep. 1984.
6. V. I. Levenshtein, "Bounds for deletion/insertion correcting codes," in Proc. IEEE Int. Symp. Inf. Theory, Lausanne, Switzerland, 2002, p. 370. (Pubitemid 34964990)
7. V. S. Pless and W. C. Huffman, Eds., Handbook of Coding Theory, Volume II, 1st ed. Amsterdam, The Netherlands: North Holland,Nov. 1998.
8. H. Mercier, V. Bhargava, and V. Tarokh, "A survey of error-correcting codes for channels with symbol synchronization errors," IEEE Commun. Surv. Tuts., vol. 12, no. 1, pp. 87-96, 2010.
9. J. Ullman, "On the capabilities of codes to correct synchronization errors," IEEE Trans. Inf. Theory, vol. 13, no. 1, pp. 95-105, Jan. 1967.
10. J. Ullman, "Near-optimal, single-synchronization-error-correcting code," IEEE Trans. Inf. Theory, vol. 12, no. 4, pp. 418-424, Oct. 1966.
11. A. Helberg and H. Ferreira, "On multiple insertion/deletion correcting codes," IEEE Trans. Inf. Theory, vol. 48, no. 1, pp. 305-308, Jan. 2002. (Pubitemid 34125331)
12. K. Abdel-Ghaffar, F. Paluni, H. Ferreira, and W. Clarke, "On Helberg's generalization of the Levenshtein code for multiple deletion/insertion error correction," IEEE Trans. Inf. Theory, vol. 58, no. 3, pp. 1804-1808, Mar. 2012.
13. L. Calabi andW. Hartnett, "Some general results of coding theory with applications to the study of codes for the correction of synchronization errors," Inf. Control, vol. 15, no. 3, pp. 235-249, Sep. 1969.
14. E. Tanaka and T. Kasai, "Synchronization and substitution error-correcting codes for the Levenshteinmetric," IEEE Trans. Inf. Theory, vol. 22, no. 2, pp. 156-162, Mar. 1976.
15. R. R. Varshamov, "A class of codes for asymmetric channels and a problem from the additive theory of numbers," IEEE Trans. Inf. Theory, vol. 19, no. 1, pp. 92-95, Jan. 1973.
16. S. Butenko, P. Pardalos, I. Sergienko, V. Shylo, and P. Stetsyuk, "Finding maximum independent sets in graphs arising from coding theory," in Proc. ACM Symp. Appl. Comput., New York, NY, USA, 2002, pp. 542-546. (Pubitemid 35009646)
17. L. Schulman and D. Zuckerman, "Asymptotically good codes correcting insertions, deletions, and transpositions," IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2552-2557, Nov. 1999. (Pubitemid 30514225)
18. F. Khajouei,M. Zolghadr, and N. Kiyavash, "An algorithmic approach for finding deletion correcting codes," in Proc. IEEE Inf. TheoryWorkshop, Paraty, Brazil, Oct. 2011, pp. 25-29.
19. D. Cullina, A. A. Kulkarni, and N. Kiyavash, "A coloring approach to constructing deletion correcting codes from constant weight subgraphs," in Proc. IEEE Int. Symp. Inf. Theory, Cambridge, USA, 2012, pp. 513-517.
20. R. Roth and P. Siegel, "Lee-metric BCH codes and their application to constrained and partial-response channels," IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1083-1096, Jul. 1994.
21. H. Hilden, D. Howe, and J. E. J.Weldon, "Shift error correcting modulation codes," IEEE Trans. Magn., vol. 27, no. 6, pp. 4600-4605, Nov. 1991.
22. A. Bours, "Construction of fixed-length insertion/deletion correcting runlength-limited codes," IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1841-1856, Nov. 1994.
23. L. Cheng, H. Ferreira, and I. Broere, "Moment balancing templates for-constrained codes and run-length limited sequences," IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 2244-2252, Apr. 2012.
24. F. Paluni, K. Abdel-Ghaffar, H. Ferreira, andW. Clarke, "A multiple insertion/deletion correcting code for run-length limited sequences," IEEE Trans. Inf. Theory, vol. 58, no. 3, pp. 1809-1824, Mar. 2012.
25. V. I. Levenshtein, "On perfect codes in deletion and insertion metric," Discrete Math. Appl., vol. 2, no. 3, pp. 241-258, Oct. 1992.
26. D. S. Hirschberg and M. Regnier, "Tight bounds on the number of string subsequences," J. Discrete Algorithms, vol. 1, no. 1, 2000.
27. T. Swart and H. Ferreira, "A note on double insertion/deletion correcting codes," IEEE Trans. Inf. Theory, vol. 49, no. 1, pp. 269-273, Jan. 2003.
28. H. Mercier, M. Khabbazian, and V. Bhargava, "On the number of subsequences when deleting symbols from a string," IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3279-3285, Jul. 2008.
29. Y. Liron and M. Langberg, "A characterization of the number of subsequences obtained via the deletion channel," in CoRR, 2012, vol. abs/1202.1644 [Online]. Available: http://arxiv.org/abs/1202.1644
30. V. I. Levenshtein, "Efficient reconstruction of sequences," IEEE Trans. Inf. Theory, vol. 47, no. 1, pp. 2-22, Jan. 2001. (Pubitemid 32248189)
31. V. I. Levenshtein, "Efficient reconstruction of sequences from their subsequences or supersequences," J. Combinatorial Theory, vol. 93, no. 2, pp. 310-332, 2001. (Pubitemid 33379343)
32. M. Mitzenmacher, "Polynomial time low-density parity-check codes with rates very close to the capacity of the-ary random deletion channel for large ," IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5496-5501, Dec. 2006. (Pubitemid 46445388)
33. Y. Kanoria and A. Montanari, On the deletion channel with small deletion probability 2009 [Online]. Available: http://arxiv.org/abs/0912.5176
34. S. Diggavi, M. Mitzenmacher, and H. D. Pfister, "Capacity upper bounds for the deletion channel," in Proc. IEEE Int. Symp. Inf. Theory, Nice, France, Jun. 2007, pp. 1716-1720.
35. D. B.West, Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, Sep. 2000.
36. Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, D. Sankoff and J. B. Kruskal, Eds. Reading, MA, USA: Addison-Wesley, 1983, Advanced Book Program.
37. C. Berge, Hypergraphs, ser. Combinatorics of Finite Sets, 1st ed. Amsterdam, The Netherlands: North Holland, Aug. 1989, vol. 45.
38. E. R. Scheinerman and D. H. Ullman, Fractional Graph Theory: A Rational Approach to the Theory of Graphs. NewYork,NY,USA: Dover, Dec. 2011.
39. A. Schrijver, Theory of Linear and Integer Programming. NewYork, NY, USA: Wiley, Jun. 1998.
40. J. Feldman, M.Wainwright, and D. Karger, "Using linear programming to decode binary linear codes," IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 954-972, Mar. 2005. (Pubitemid 40377560)
41. Z. Füredi, "Maximum degree and fractional matchings in uniform hypergraphs," Combinatorica, vol. 1, no. 2, pp. 155-162, 1981.
42. R. Aharoni, R. Holzman, andM. Krivelevich, "On a theorem of Lovász on covers in-partite hypergraphs," Combinatorica, vol. 16, no. 2, pp. 149-174, Jun. 1996.
43. R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge, U.K.: Cambridge Univ. Press, 1995.
44. B. Marcus, P. Siegel, and R. Roth, "An introduction to coding for constrained systems," in Handbook of Coding Theory, W. C. Huffman and V. Pless, Eds. New York, NY, USA: Elsevier, 1998.
45. B. Marcus, P. Siegel, and R. Roth, An introduction to coding for constrained systems 2001 [Online]. Available: http://www.math.ubc.ca/~marcus/ Handbook/index.html