Constant-rank codes and their connection to constant-dimension codes

IEEE Transactions on Information Theory

Volumn 56, Issue 7, 2010, Pages 3207-3216

  Gadouleau, Maximilien ^a   Yan, Zhiyuan ^b  

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

^b Lehigh University   (United States)

Author keywords

Constant dimension codes; constant weight codes; Error control codes; Injection metric; Network coding; Random linear network coding; Rank metric codes; Subspace codes; Subspace metric

Indexed keywords

ERROR CONTROL CODE; LINEAR NETWORK CODING; NETWORK CODING; RANK METRIC CODES; SUBSPACE CODES; SUBSPACE METRIC;

ASYMPTOTIC ANALYSIS; INFORMATION THEORY; LINEAR NETWORKS; OPTIMIZATION; SIGNAL ENCODING;

OPTIMAL SYSTEMS;

EID: 77953775753     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2010.2048447     Document Type: Article

References (42)

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," presented at the Allerton Conf. on Comm., Control, and Computing, Monticello, IL, Oct. 2003.
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. (Pubitemid 46445282)
4. N. Cai and R. W. Yeung, "Network coding and error correction," in Proc. IEEE Information Theory Workshop, Bangalore, India, Oct. 2002, pp. 20-25.
5. L. Song, R. W. Yeung, and N. Cai, "Zero-error network coding for acyclic networks," IEEE Trans. Inf. Theory, vol.49, no.12, pp. 3129-3139, Dec. 2003.
6. 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.
7. N. Cai and R. W. Yeung, "Network error correction, part II: Lower bounds," Commun. Inf. Syst., vol.6, no.1, pp. 37-54, 2006.
8. Z. Zhang, "Linear network error correction codes in packet networks," IEEE Trans. Inf. Theory, vol.54, no.1, pp. 209-218, Jan. 2008.
9. R. Koetter 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. D. Silva, F. R. Kschischang, and R. Koetter, "A rank-metric approach to error control in random network coding," IEEE Trans. Inf. Theory, vol.54, no.9, pp. 3951-3967, Sep. 2008.
11. S. Jaggi, M. Langberg, S. Katti, T. Ho, and D. K. Médard, "Resilient network coding in the presence of byzantine adversaries," in Proc. IEEE Infocom, Anchorage, AK, May 2007, pp. 616-624.
12. P. Delsarte, "Association schemes and -designs in regular semilattices," J. Combin. Theory A, vol.20, no.2, pp. 230-243, Mar. 1976.
13. L. Chihara, "On the zeros of the askey-wilson polynomials with applications to coding theory," SIAM J. Math. Anal., vol.8, no.1, pp. 191-207, Jan. 1987.
14. "On metrics for error correction in network coding," IEEE Trans. Inf. Theory, vol.55, no.10, pp. 5479-5490, Dec. 2009.
15. T. Etzion and A. Vardy, "Error-correcting codes in projective space," in Proc. IEEE Int. Symp. Information Theory, Toronto, ON, Canada, Jul. 2008, pp. 871-875.
16. E. M. Gabidulin and M. Bossert, "Codes for network coding," in Proc. IEEE Int. Symp. on Information Theory, Toronto, ON, Canada, Jul. 2008, pp. 867-870.
17. W. J. Martin and X. J. Zhu, "Anticodes for the Grassmann and bilinear forms graphs," Designs, Codes, Cryptogr., vol.6, no.1, pp. 73-79, Jul. 1995.
18. R. Ahlswede, H. K.Aydinian, and L. H. Khachatrian, "On perfect codes and related concepts," Designs, Codes, Cryptogr., vol.22, no.3, pp. 221-237, Jan. 2001.
19. M. Schwartz and T. Etzion, "Codes and anticodes in the Grassmann graph," J. Combin. Theory Ser. A, vol.97, no.1, pp. 27-42, Jan. 2002.
20. S.-T. Xia and F.-W. Fu, "Johnson type bounds on constant dimension codes," Designs, Codes, Cryptogr., vol.50, no.2, pp. 163-172, Feb. 2009.
21. P. Frankl and R. M. Wilson, "The Erdös-Ko-Rado theorem for vector spaces," J. Combin. Theory A, vol.43, no.2, pp. 228-236, Nov. 1986.
22. T. Etzion, "Perfect byte-correcting codes," IEEE Trans. Inf. Theory, vol.44, no.7, pp. 3140-3146, Nov. 1998.
23. H. Wang, C. Xing, and R. Safani-Naini, "Linear authentication codes: Bounds and constructions," IEEE Trans. Inf. Theory, vol.49, no.4, pp. 866-872, Apr. 2003.
24. V. Skachek, "Recursive code construction for random networks," IEEE Trans. Inf. Theory, vol.56, no.3, pp. 1378-1382, Mar. 2010.
25. T. Etzion and N. Silberstein, "Error-correcting codes in projective spaces via rank-metric codes and ferrers diagrams," IEEE Trans. Inf. Theory, vol.55, no.7, pp. 2909-2919, Jul. 2009.
26. P. Delsarte, "Bilinear forms over a finite field, with applications to coding theory," J. Combin. Theory A, vol.25, no.3, pp. 226-241, Nov. 1978.
27. E. M. Gabidulin, "Theory of codes with maximum rank distance," Probl. Inf. Transmission, vol.21, no.1, pp. 1-12, Jan. 1985.
28. R. M. Roth, "Maximum-rank array codes and their application to crisscross error correction," IEEE Trans. Inf. Theory, vol.37, no.2, pp. 328-336, Mar. 1991.
29. E. M. Gabidulin, A. V. Paramonov, and O. V. Tretjakov, "Ideals over a non-commutative ring and their application in cryptology," in Proc. Eurocrypt, Brighton, U.K., Apr. 1991, pp. 482-489.
30. P. Lusina, E. M. Gabidulin, and M. Bossert, "Maximum rank distance codes as space-time codes," IEEE Trans. Inf. Theory, vol.49, no.10, pp. 2757-2760, Oct. 2003.
31. G. E. Andrews, The Theory of Partitions, ser. Encyclopedia of Mathematics and its Applications, G.-C. Rota, Ed. Reading, MA: Addison- Wesley, 1976, vol.2.
32. M. Gadouleau and Z. Yan, "On the decoder error probability of bounded rank-distance decoders for maximum rank distance codes," IEEE Trans. Inf. Theory, vol.54, no.7, pp. 3202-3206, Jul. 2008.
33. M. Gadouleau and Z. Yan, "Bounds on covering codes with the rank metric," IEEE Commun. Lett., vol.13, pp. 691-693, Sep. 2009.
34. A. Kshevetskiy and E. M. Gabidulin, "The new construction of rank codes," in Proc. IEEE Int. Symp. Inf. Theory, Adelaide, Australia, Sep. 2005, pp. 2105-2108.
35. A. Kohnert and S. Kurz, "Construction of large constant dimension codes with a prescribed minimum distance," Mathematical Meth. Comput. Sci., LNCS, vol.5393, pp. 31-42, Dec. 2008.
36. A. R. Rao and P. Bhimasankaram, Linear Algebra, 2nd ed. Guragon, India: Hindustan Book Agency, 2000.
37. G. Marsaglia and G. P. H. Styan, "Equalities and inequalities for ranks of matrices," Linear and Multilinear Algebra, vol.2, pp. 269-292, 1974.
38. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.k.: Cambridge Univ. Press, 1985.
39. M. Gadouleau and Z. Yan, "Construction and covering properties of constant-dimension codes," IEEE Trans. Inf. Theory, submitted for publication.
40. S. M. Johnson, "A new upper bound for error-correcting codes," IRE Trans. Inf. Theory, vol.8, no.3, pp. 203-207, Apr. 1962.
41. L. A. Bassalygo, "Newupper bounds for error correcting codes," Probl. Inf. Transmission, vol.1, no.4, pp. 32-35, Oct.-Dec. 1965.
42. M. Gadouleau and Z. Yan, "Packing and covering properties of rank metric codes," IEEE Trans. Inf. Theory, vol.54, no.9, pp. 3873-3883, Sep. 2008.