A rank-metric approach to error control in random network coding

IEEE Transactions on Information Theory

Volumn 54, Issue 9, 2008, Pages 3951-3967

  Silva, Danilo ^a   Kschischang, Frank R ^a   Köetter, Ralf ^b  

^a UNIVERSITY OF TORONTO   (Canada)

^b TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Constant dimension codes; Error correction; Linearized polynomials; Random network coding; Rank metric codes

Indexed keywords

COMPUTER NETWORKS; DECODING; ERROR ANALYSIS; MATRIX ALGEBRA; PROGRAMMING THEORY; SIGNAL ENCODING;

CONSTANT-DIMENSION CODES; LINEARIZED POLYNOMIALS; METRIC CODES; RANDOM NETWORK CODING; RANK-METRIC CODES;

ERROR CORRECTION;

EID: 51349146209     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2008.928291     Document Type: Article

References (29)

1. T. Ho, R. Koetter, M. Médard, D. R. Karger, and M. Effros, "The benefits of coding over routing in a randomized setting," in Proc. IEEE Int. Symp. Information Theory, Yokohama, Japan, Jun./Jul. 2003, p. 442.
2. P. A. Chou, Y. Wu, and K. Jain, "Practical network coding," in Proc. Allerton Conf. on Communications, Control, and Computing, Monticello, IL, Oct. 2003, pp. 40-49.
3. T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi, and B. Leong, "A random linear network coding approach to multicast," IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4413-4430, Oct. 2006.
4. N. Cai and R. W. Yeung, "Network coding and error correction," in Proc. 2002 IEEE Information Theory Workshop, Bangalore, India, Oct. 2002, pp. 119-122.
5. R. W. Yeung and N. Cai, "Network error correction, part I: Basic concepts and upper bounds," Commun. Inf. Syst., vol. 6, no. 1, pp. 19-36, 2006.
6. N. Cai and R. W. Yeung, "Network error correction, Part II: Lower bounds," Commun. Inf. Syst., vol. 6, no. 1, pp. 37-54, 2006.
7. Z. Zhang, "Linear network error correction codes in packet networks," IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 209-218, Jan. 2008.
8. S. Jaggi, M. Langberg, S. Katti, T. H. D. Katabi, and M. Médard, "Resilient network coding in the presence of Byzantine adversaries," in Proc. 26th IEEE Int. Conf. Computer Communication, Anchorage, AK, May 2007, pp. 616-624.
9. R. Kötter and F. R. Kschischang, "Coding for errors and erasures in random network coding," IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3579-3591, Aug. 2008.
10. P. Delsarte, "Bilinear forms over a finite field, with applications to coding theory," J. Comb. Theory. Ser. A, vol. 25, pp. 226-241, 1978.
11. E. M. Gabidulin, "Theory of codes with maximum rank distance," Probl. Inf. Transm., vol. 21, no. 1, pp. 1-12, 1985.
12. R. M. Roth, "Maximum-rank array codes and their application to criss-cross error correction," IEEE Trans. Inf. Theory, vol. 37, no. 2, pp. 328-336, Mar. 1991.
13. G. Richter and S. Plass, "Error and erasure decoding of rank-codes with a modified Berlekamp-Massey algorithm," in Proc. ITG Conf. Source and Channel Coding, Erlangen, Germany, Jan. 2004, pp. 249-256.
14. P. Loidreau, "A Welch-Berlekamp like algorithm for decoding Gabidulin codes," in Proc. 4th Int. Workshop on Coding and Cryptography, Bergen, Norway, Mar. 2005, pp. 36-45.
15. E. M. Gabidulin, A. V. Paramonov, and O. V. Tretjakov, "Rank errors and rank erasures correction," in Proc. 4th Int. Colloq. Coding Theory, Dilijan, Armenia, 1991, Yerevan, Armenia, 1992, pp. 11-19.
16. E. M. Gabidulin and N. I. Pilipchuk, "A new method of erasure correction by rank codes," in Proc. IEEE Int. Symp. Information Theory, Yokohama, Japan, Jun./Ju. 2003, pp. 423-423.
17. G. Richter and S. Plass, "Fast decoding of rank-codes with rank errors and column erasures," in Proc. IEEE Int. Symp. Information Theory Chicago, IL, Jun./Jul. 2004, p. 398.
18. N. I. Pilipchuk and E. M. Gabidulin, "Error and erasure correcting algorithms for rank codes," Des., Codes, Cryptogr., to be published.
19. R. M. Roth and G. Seroussi, "Location-correcting codes," IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 554-565, Mar. 1996.
20. P. Loidreau, "Étude et optimization de cryptosystèmesà clé publique fondés sur la théorie des codes correcteurs," Ph.D. dissertation, École Polytechnique, Paris, France, 2001.
21. M. Gadouleau and Z. Yan, "Properties of codes with the rank metric," in Proc. IEEE Globecom 2006, San Francisco, CA, Nov. 1, 2006.
22. H. Wang, C. Xing, and R. Safavi-Naini, "Linear authentication codes: Bounds and constructions," IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 866-872, Apr. 2003.
23. S.-T. Xia and F.-W. Fu, "Johnson type bounds on constant dimension codes," 2007. [Online]. Available: http://arxiv.org/abs/0709.1074, submitted for publication
24. D. Silva and F. R. Kschischang, "On metrics for error correction in network coding," 2008. [Online]. Available: http://arxiv.org/abs/ 0805.3824, submitted for publication
25. R. Lidl and H. Niederreiter, Finite Fields. Reading, MA: Addison-Wesley, 1983.
26. F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes Amsterdam, The Netherlands: North-Holland, 1977.
27. V. Skachek and R. M. Roth, "Probabilistic algorithm for finding roots of linearized polynomials," Des., Codes and Cryptography, vol. 46, no. 1, pp. 17-23, 2008.
28. E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
29. D. Silva and F. R. Kschischang, "Rank-metric codes for priority encoding transmission with network coding," in Proc. 10th Canadian Workshop on Information Theory, Edmonton, AB, Canada, Jun. 2007, pp. 81-84.