Using linear programming to decode binary linear codes

IEEE Transactions on Information Theory

Volumn 51, Issue 3, 2005, Pages 954-972

  Feldman, Jon ^a   Wainwright, Martin J ^b   Karger, David R ^c  

^a Och Spine at New York Presbyterian Hospitals   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Belief propagation (BP); Iterative decoding; Linear codes; Linear programming (LP); Low density parity check (LDPC) codes; LP decoding; Minimum distance; Pseudocodewords

Indexed keywords

ALGORITHMS; CODES (SYMBOLS); DECODING; GRAPH THEORY; ITERATIVE METHODS; LINEAR PROGRAMMING; MATRIX ALGEBRA; THEOREM PROVING;

BELIEF PROPAGATION (BP); ITERATIVE DECODING; LINEAR CODES; LOW DENSITY PARITY CHECK (LDPC) CODES; LP DECODING; MINIMUM DISTANCE; PSEUDOCODEWORDS;

COMMUNICATION CHANNELS (INFORMATION THEORY);

EID: 15044355718     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2004.842696     Document Type: Article

References (37)

1. J. Feldman and D. R. Karger, "Decoding turbo-like codes via linear programming," in Proc. 43rd Annu. IEEE Symp. Foundations of Computer Science (FOCS), Vancouver, BC, Canada, Nov. 2002, pp. 251-260.
2. J. Feldman, M. J. Wainwright, and D. R. Karger, "Linear programming-based decoding of turbo-like codes and its relation to iterative approaches," in Proc. Allerton Conf. Communications, Control and Computing, Monticello, IL, Oct. 2002.
3. C. Di, D. Proietti, I. E. Telatar, T. J. Richardson, and R. L. Urbanke, "Finite-length analysis of low-density parity-check codes on the binary erasure channels," IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1570-1579, Jun. 2002.
4. N. Wiberg, "Codes and decoding on general graphs," Ph.D. dissertation, Linköping University, Linköping, Sweden, 1996.
5. G. D. Forney, F. R. Kschischang, B. Marcus, and S. Tuncel, "Iterative decoding of tail-biting trellises and connections with symbolic dynamics," in Codes. Systems and Graphical Models. New York: Springer-Verlag, 2001, pp. 239-241.
6. G. D. Forney, R. Koetter, F. R. Kschischang, and A. Reznik, "On the effective weights of pseudocodewords for codes defined on graphs with cycles," in Codes, Systems and Graphical Models. New York: Springer-Verlag, 2001, pp. 101-112.
7. R. Gallager, "Low-density parity-check codes," IRE Trans. Inf. Theory, vol. IT-8, no. 1, pp. 21-28, Jan. 1962.
8. D. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399-431, Mar. 1999.
9. M. Sipser and D. Spielman, "Expander codes," IEEE Trans. Inf. Theory, vol 42, no. 6, pp. 1710-1722, Nov. 1996.
10. S.-Y. Chung, G. D. Forney, T. Richardson, and R. Urbanke, "On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit," IEEE Commun. Lett., vol. 5, no. 2, pp. 58-60, Feb. 2001.
11. R. McEliece, D. MacKay, and J. Cheng, "Turbo decoding as an instance of Pearl's belief propagation algorithm," IEEE J. Sel. Areas Commun. vol 16, no. 2, pp. 140-152, Feb. 1998.
12. T. J. Richardson and R. L. Urbanke, "The capacity of low-density parity-check codes under message-passing decoding," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 599-618, Feb. 2001.
13. M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, "Improved low-density parity-check codes using irregular graphs and belief propagation," in Proc. IEEE Int. Symp. Information Theory, Cambridge, MA, Oct. 1998, p. 117.
14. B. J. Frey, R. Koetter, and A. Vardy, "Signal-space characterization of iterative decoding," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 766-781, Feb. 2001.
15. J. Feldman, M. J. Wainwright, and D. R. Karger, "Using linear programming to decode linear codes," presented at the 37th Annu. Conf. on Information Sciences and Systems (CISS '03), Baltimore, MD, Mar. 2003.
16. J. Feldman, D. R. Karger, and M. J. Wainwright, "LP decoding," in Proc. 41st Annu. Allerton Conf. Communications, Control, and Computing Monticello, IL, Oct. 2003.
17. J. Feldman, "Decoding error-correcting codes via linear programming," Ph.D. dissertation, MIT, Cambridge, MA, 2003.
18. J. Feldman, T. Malkin, R. A. Servedio, C. Stein, and M. J. Wainwright, "LP decoding corrects a constant fraction of errors," in Proc. IEEE Int. Symp. Information Theory, Chicago, IL, Jun./Jul. 2004, p. 68.
19. J. Feldman and C. Stein, "LP decoding achieves capacity," in Proc. Symp. Discrete Algorithms (SODA '05), Vancouver, BC, Canada, Jan. 2005.
20. R. Koetter and P. O. Vontobel, "Graph-covers and iterative decoding of finite length codes," in Proc. 3rd Int. Symp. Turbo Codes, Brest, France, Sep. 2003, pp. 75-82.
21. P. Vontobel and R. Koetter, "On the relationship between linear programming decoding and max-product decoding," paper submitted to Int. Symp. Information Theory and its Applications, Parma, Italy, Oct. 2004.
22. P. Vontobel, R. Koetter, "Lower bounds on the minimum pseudo-weight of linear codes," in Proc. IEEE Int. Symp. Information Theory, Chicago, IL, Jun./Jul. 2004, p. 70.
23. A. Schrijver, Theory of Linear and Integer Programming. New York: Wiley, 1987.
24. M. Grotschel, L. Loväsz, and A. Schrijver, "The ellipsoid method and its consequences in combinatorial optimization," Combinatorica, vol. 1, no. 2, pp. 169-197, 1981.
25. E. Berlekamp, R. J. McEliece, and H. van Tilborg, "On the inherent intractability of certain coding problems," IEEE Trans. Inf. Theory vol. IT-24, no. 3, pp. 384-386, May 1978.
26. M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky, "MAP estimation via agreement on (hyper)trees: Message-passing and linear programming approaches," in Proc. 40th Annu. Allerton Conf. Communication, Control, and Computing, Monticello, IL, Oct. 2002.
27. J. S. Yedidia, W. T. Freeman, and Y. Weiss, "Understanding Belief Propagation and its Generalizations," Mitsubishi Electric Res. Labs, Tech. Rep. TR2001-22, 2002.
28. D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization Belmont, MA: Athena Scientific, 1997.
29. R. G. Jeroslow, "On defining sets of vertices of the hypercube by linear inequalities," Discr. Math., vol. 11, pp. 119-124, 1975.
30. M. Yannakakis, "Expressing combinatorial optimization problems by linear programs," J. Comp. Syst. Sci., vol. 43, no. 3, pp. 441-466, 1991.
31. G. D. Forney Jr., "Codes on graphs: Normal realizations," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 529-548, Feb. 2001.
32. R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory, vol. IT-27, no. 5, pp. 533-547, Sep. 1981.
33. J. Rosenthal and P. O. Vontobel, "Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis," in Proc. 38th Annu. Allerton Conf. Communication, Control, and Computing, Monticello, IL, Oct. 2000, pp. 248-257.
34. D. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, 1995.
35. L. Loväsz and A. Schrijver, "Cones of matrices and set-functions and 0-1 optimization," SIAM J. Optimiz., vol. 1, no. 2, pp. 166-190, 1991.
36. H. D. Sherali and W. P. Adams, "A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems," SIAM J. Optimiz., vol. 3, pp. 411-430, 1990.
37. User's Manual for ILOG CPLEX, 7.1 ed., ILOG, Inc., Mountain View, CA, 2001.