Spatially coupled ensembles universally achieve capacity under belief propagation

IEEE Transactions on Information Theory

Volumn 59, Issue 12, 2013, Pages 7761-7813

  Kudekar, Shrinivas ^a   Richardson, Tom ^a   Urbanke, Rudiger L ^b  

^a QUALCOMM INC   (United States)

^b EPFL   (Switzerland)

Author keywords

Belief propagation (BP); capacity achieving codes; channel coding; convolutional low density parity check (LDPC) codes; iterative decoding; LDPC codes; spatial coupling; spatially coupled codes; threshold saturation

Indexed keywords

BELIEF PROPAGATION; CAPACITY-ACHIEVING CODES; COUPLED CODE; LDPC CODES; LOW-DENSITY PARITY-CHECK (LDPC) CODES; SPATIAL COUPLINGS; THRESHOLD SATURATIONS;

CHANNEL CODING; FORWARD ERROR CORRECTION; ITERATIVE DECODING;

CONVOLUTION;

EID: 84889590124     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2013.2280915     Document Type: Article

References (121)

1. C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J., vol. 27, pp. 379-423, Jul./Oct. 1948.
2. R. W. Hamming, "Error detecting and error correcting codes," Bell Syst. Tech. J., vol. 26, no. 2, pp. 147-160, 1950.
3. M. J. E. Golay, "Notes on digital coding," Proc. IRE, vol. 37, pp. 657-657, Jun. 1949.
4. G. Forney and D. Costello, "Channel coding: The road to channel capacity," Proc. IEEE, vol. 95, no. 6, pp. 1150-1177, Jun. 2007.
5. A. Hocquenghem, "Codes correcteurs d'erreurs," Chiffres, vol. 2, pp. 147-156, 1959.
6. R. C. Bose and D. K. Ray-Chaudhuri, "On a class of error-correcting binary group codes," Inf. Control, vol. 3, pp. 68-79, Mar. 1960.
7. I. S. Reed and G. Solomon, "Polynomial codes over certain finite fields," SIAM J., vol. 8, no. 2, pp. 300-304, Jun. 1960.
8. E. R. Berlekamp, Algebraic Coding Theory. Walnut Creek, CA, USA: Aegean Park Press, 1984.
9. J. Massey, "Shift-register synthesis and BCH decoding," IEEE Trans. Inf. Theory, vol. 15, no. 1, pp. 122-127, Jan. 1969.
10. M. Sudan, "Decoding Reed-Solomon codes beyond the error-correction diameter," presented at the Allerton Conf. Commun., Control, and Computing, Monticello, IL, USA, 1997.
11. V. Guruswami and M. Sudan, "Improved decoding of Reed-Solomon and algebraic-geometry codes," IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 1757-1767, Sep. 1999. (Pubitemid 32871911)
12. R. Kötter and A. Vardy, "Algebraic soft-decision decoding of Reed-Solomon codes," IEEE Trans. Inf. Theory, vol. 49, no. 11, pp. 2809-2825, Nov. 2003.
13. P. Elias, "Coding for noisy channels," IRE Int. Convent. Record, pp. 37-46, Mar. 1955.
14. J. M.Wozencraft, Sequential decoding for reliable communication Research Lab. Electron, MIT, Cambridge, MA, USA, 1957, Tech. Rep. 325.
15. A. J. Viterbi, "Error bounds of convolutional codes and an asymptotically optimum decoding algorithm," IEEE Trans. Inf. Theory, vol. 13, no. 2, pp. 260-269, Apr. 1967.
16. G. D. Forney, Jr., Review of random tree codes NASA Ames Res. Centre,, Contract NAS2-3637, NASA CR73176, Dec. 1967.
17. J. K. Omura, "On the Viterbi decoding algorithm," IEEE Trans. Inf. Theory, vol. IT-15, no. 1, pp. 177-179, Jan. 1969.
18. J. A. Heller, Short constraint length convolutional codes Jet Propulsion Lab., California Institute of Technology, Pasadena, CA, USA, Space Prog. Summary 37-54, 1968.
19. J. A. Heller, Improved performance of short constraint length convolutional codes Jet Propulsion Lab., California Institute of Technology, Pasadena, CA, USA, Space Prog. Summary 37-56, 1969.
20. R. de Buda, "The upper error bound of a new near-optimal code," IEEE Trans. Inf. Theory, vol. IT-21, no. 4, pp. 441-445, Jul. 1975.
21. R. de Buda, "Some optimal codes have structure," IEEE J. Sel. Areas Commun., vol. 7, no. 6, pp. 893-899, Aug. 1989.
22. H. A. Loeliger, "Averaging bounds for lattice and linear codes," IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1767-1773, Nov. 1997.
23. R. Urbanke and B. Rimoldi, "Lattice codes can achieve capacity on the AWGN channel," IEEE Trans. Inf. Theory, vol. 44, no. 1, pp. 273-278, Jan. 1998. (Pubitemid 128737895)
24. U. Erez and R. Zamir, "Achieving 1/2 log(1+snr) on the AWGN channel with lattice encoding and decoding," IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2293-2314, Oct. 2004.
25. G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory, vol. IT-28, no. 1, pp. 55-67, Jan. 1982.
26. G. Ungerboeck, "Trellis-coded modulation with redundant signal sets - Part I: Introduction," IEEE Commun. Mag., vol. CM-25, no. 2, pp. 5-11, Feb. 1987. (Pubitemid 17549477)
27. G. Ungerboeck, "Trellis-coded modulation with redundant signal sets - Part II: State of the art," IEEE Commun. Mag., vol. CM-25, no. 2, pp. 12-21, Feb. 1987. (Pubitemid 17549476)
28. G. D. Forney, Jr., "Coset codes - Part I: Introduction and geometrical classification," IEEE Trans. Inf. Theory, vol. 34, no. 5, pp. 1123-1151, Sep. 1988.
29. G. D. Forney, Jr., "Coset codes - Part I: Introduction and geometrical classification," IEEE Trans. Inf. Theory, vol. 34, no. 5, pp. 1152-1187, Sep. 1988.
30. C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding," in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, May 1993, pp. 1064-1070. (Pubitemid 23674800)
31. D. A. Spielman, "Linear-time encodeable and decodable error-correcting codes," IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 1723-1731, Nov. 1996. (Pubitemid 126769171)
32. M. Sipser and D. A. Spielman, "Expander codes," IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 1710-1722, Nov. 1996. (Pubitemid 126769170)
33. D. J. C. MacKay and R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett., vol. 32, no. 18, pp. 1645-1646, Aug. 1996. (Pubitemid 126546283)
34. D. J. C. MacKay and R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett., vol. 33, no. 6, pp. 457-458, Mar. 1997. (Pubitemid 127587405)
35. D. J. C. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399-431, Mar. 1999
36. [Online]. Available: http://www.inference. phy.cam.ac.uk/mackay/ abstracts/mncN.html
37. D. J. C. MacKay and R. M. Neal, C. Boyd, Ed., "Good codes based on very sparse matrices," in Proc. Cryptography Coding, Berlin, Germany, 1995, pp. 100-111. (Pubitemid 126019851)
38. R. G. Gallager, Low-Density Parity-Check Codes. Cambridge,MA, USA: MIT Press, 1963.
39. N. Wiberg, "Codes and decoding on general graphs," Ph.D. dissertation, Linköping University, Linköping, Sweden, 1996.
40. F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, "Factor graphs and the sum-product algorithm," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498-519, Feb. 2001. (Pubitemid 32318087)
41. M. Luby, M. Mitzenmacher, A. Shokrollahi, and D. A. Spielman, "Improved low-density parity-check codes using irregular graphs," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 585-598, Feb. 2001. (Pubitemid 32318091)
42. M. Luby, M. Mitzenmacher, A. Shokrollahi, and D. A. Spielman, "Analysis of low density codes and improved designs using irregular graphs," in Proc. 30th Annu. ACM Symp. Theory Comput., 1998, pp. 249-258.
43. S.-Y. Chung, T. Richardson, and R. Urbanke, "Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 657-670, Feb. 2001. (Pubitemid 32318095)
44. M. Luby, M. Mitzenmacher, A. Shokrollahi, and D. A. Spielman, "Efficient erasure correcting codes," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 569-584, Feb. 2001. (Pubitemid 32318090)
45. M. Luby, M. Mitzenmacher, A. Shokrollahi, D. A. Spielman, and V. Stemann, "Practical loss-resilient codes," in Proc. 29th Annu. ACM Symp. Theory Comput., 1997, pp. 150-159.
46. T. Richardson and R. 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. (Pubitemid 32318092)
47. T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge, U.K.: Cambridge Univ. Press, 2008.
48. D. Divsalar, H. Jin, and R. J. McEliece, "Coding theorems for turbolike codes," in Proc. Allerton Conf. Commun., Control, Comput.,Monticello, IL, USA, Sep. 1998, pp. 201-210.
49. J. Thorpe, K. Andrews, and S. Dolinar, "Methodologies for designing LDPC codes using protographs and circulants," in Proc. IEEE Int. Symp. Inf. Theory, Chicago, IL, USA, Jun. 2004, pp. 238-238.
50. D. Divsalar, C. R. Jones, S. Dolinar, and J. Thorpe, "Protograph based LDPC codeswithminimumdistance linearly growing with block size," in Proc. Global Telecommun. Conf., Nov. 2005, pp. 1152-1156. (Pubitemid 46171310)
51. E. Arikan, "Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels," IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051-3073, Jul. 2009.
52. S. Hassani, K. Alishahi, and R. Urbanke, "On the scaling of polar codes - Part II: The behavior of un-polarized channels," in Proc. IEEE Int. Symp. Inf. Theory, Jun. 2010, pp. 879-883.
53. S. Korada, A. Montanari, E. Telatar, and R. Urbanke, "An empirical scaling law for polar codes," in Proc. IEEE Int. Symp. Inf. Theory, Jun. 2010, pp. 884-888.
54. A. Goli, S. Hassani, and R. Urbanke, "Universal bounds on the scaling behavior of polar codes," in Proc. IEEE Int. Symp. Inf. Theory, Jul. 2012, pp. 1957-1961.
55. S. Kudekar, T. Richardson, and R. Urbanke, "Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC," IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803-834, Feb. 2011.
56. A. J. Felström and K. S. Zigangirov, "Time-varying periodic convolutional codes with low-density parity-check matrix," IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 2181-2190, Sep. 1999.
57. S. Kudekar and H. D. Pfister, The effect of spatial coupling on compressive sensing, presented at the Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, 2010.
58. Y. Sato, K. Kasai, and K. Sakaniwa, Compressed sensing via nodebased verification recovery with band sparsematrices, presented at the 34th Symp. Inf. Theory Applications, Ousyuku, Iwate, Japan, 2011.
59. F. Krzakala, M. Mézard, F. Sausset, Y. F. Sun, and L. Zdeborová, "Statistical-physics-based reconstruction in compressed sensing," Phys. Rev. X, vol. 2, p. 021005, May 2012
60. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevX.2.021005
61. D. Donoho, A. Javanmard, and A. Montanari, Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing, CoRR vol. abs/1112.0708, 2011.
62. K. Engdahl and K. S. Zigangirov, "On the theory of low density convolutional codes I," Probl. Peredachi Inf., vol. 35, no. 4, pp. 295-310, 1999.
63. K. Engdahl, M. Lentmaier, and K. S. Zigangirov, "On the theory of low-density convolutional codes," in Proc. 13th Int. Symp. Appl. Algebra, Algorithm Error-Correcting Codes, London, U.K., 1999, pp. 77-86.
64. M. Lentmaier, D. V. Truhachev, and K. S. Zigangirov, "To the theory of low-density convolutional codes - Part II," Probl. Inf. Transmiss., no. 4, pp. 288-306, 2001.
65. R. M. Tanner,D. Sridhara, A. Sridhara, T. E. Fuja, and D. J. Costello, Jr., "LDPC block and convolutional codes based on circulant matrices," IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 2966-2984, Dec. 2004.
66. A. Sridharan, M. Lentmaier, D. J. Costello, Jr., and K. S. Zigangirov, "Convergence analysis of a class of LDPC convolutional codes for the erasure channel," presented at the Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, Oct. 2004.
67. M. Lentmaier, A. Sridharan, K. S. Zigangirov, and D. J. Costello, Jr., "Iterative decoding threshold analysis for LDPC convolutional codes," IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 5274-5289, Oct. 2010.
68. M. Lentmaier, A. Sridharan, K. S. Zigangirov, and D. J. Costello, Jr., "Terminated LDPC convolutional codes with thresholds close to capacity," CoRR vol. abs/cs/0508030, 2005.
69. M. Lentmaier and G. P. Fettweis, "On the thresholds of generalized LDPC convolutional codes based on protographs," in Proc. IEEE Int. Symp. Inf. Theory, Austin, TX, USA, Jun. 2010, pp. 709-713.
70. D. G. M. Mitchell, A. E. Pusane, K. S. Zigangirov, and D. J. Costello, Jr., "Asymptotically good LDPC convolutional codes based on protographs," in Proc. IEEE Int. Symp. Inf. Theory, Toronto, CA, USA, Jul. 2008, pp. 1030-1034.
71. M. Lentmaier, G. P. Fettweis, K. S. Zigangirov, and D. J. Costello, Jr., "Approaching capacity with asymptotically regular LDPC codes," in Proc. Inf. Theory Appl. Workshop, San Diego, CA, USA, Feb. 13, 2009, pp. 173-177.
72. A. Pusane, R. Smarandache, P. Vontobel, and J. D. J. Costello, "Deriving good LDPC convolutional codes from LDPC block codes," IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2577-2598, Feb. 2011.
73. R. Smarandache, A. Pusane, P. Vontobel, and J. D. J. Costello, "Pseudo-codewords in LDPC convolutional codes," in Proc. IEEE Int. Symp. Inf. Theory, Seattle, WA, USA, Jul. 2006, pp. 1364-1368. (Pubitemid 351244320)
74. R. Smarandache, A. Pusane, P. Vontobel, and J. D. J. Costello, "Pseudocodeword performance analysis for LDPC convolutional codes," IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2577-2598, Jun. 2009.
75. M. Papaleo, A. Iyengar, P. Siegel, J.Wolf, and G. Corazza, "Windowed erasure decoding of LDPC convolutional codes," in Proc. IEEE Inf. Theory Workshop, Cairo, Egypt, Jan. 2010, pp. 78-82.
76. A. Iyengar, M. Papaleo, P. Siegel, J. Wolf, A. Vanelli-Coralli, and G. Corazza, "Windowed decoding of protograph-based LDPC convolutional codes over erasure channels," IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 2303-2320, Apr. 2011.
77. A. Iyengar, P. Siegel, R. Urbanke, and J. Wolf, "Windowed decoding of spatially coupled codes," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 2552-2556.
78. P. Olmos and R. Urbanke, "Scaling behavior of convolutional LDPC ensembles over the BEC," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 1816-1820.
79. D. Divsalar, S. Dolinar, and C. Jones, "Constructions of protograph LDPC codes with linear minimum distance," presented at the IEEE Int. Symp. Inf. Theory, Seattle, WA, USA, Jul. 2006.
80. C. Méasson, A. Montanari, T. Richardson, and R. Urbanke, "The generalized area theorem and some of its consequences," IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 4793-4821, Nov. 2009.
81. S. Kudekar, C. Measson, T. Richardson, and R. Urbanke, Threshold saturation on BMS channels via spatial coupling, presented at the Int. Conf. Turbo Codes Related Topics, Sep. 2010.
82. M. Lentmaier, D. G. M. Mitchell, G. P. Fettweis, and D. J. Costello, Jr., Asymptotically good LDPC convolutional codes with AWGN channel thresholds close to the Shannon limit, in Proc. Sixth Int. Symp. Turbo Codes Iterative Inf. Process., Brest, France, Sep. 2010.
83. K. Kasai and K. Sakaniwa, "Spatially-coupled MacKay-Neal codes and Hsu-Anastasopoulos codes," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 747-751.
84. V. Aref and R. Urbanke, "Universal rateless codes from coupled LT codes," in Proc. IEEE Inf. Theory Workshop, Oct. 2011, pp. 277-281.
85. S. Kudekar and K. Kasai, "Threshold saturation on channels with memory via spatial coupling," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 2562-2566.
86. S. Kudekar and K. Kasai, "Spatially coupled codes over the multiple access channel," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 2816-2820.
87. K. Takeuchi, T. Tanaka, and T. Kawabata, "Improvement of BP-based CDMA multiuser detection by spatial coupling," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 1489-1493.
88. H. Uchikawa, K. Kasai, and K. Sakaniwa, "Spatially coupled LDPC codes for decode-and-forward in erasure relay channel," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 1474-1478.
89. A. Yedla, H. Pfister, and K. Narayanan, "Universality for the noisy Slepian-Wolf problem via spatial coupling," in Proc. IEEE Int. Symp. Inf. Theory, Aug. 2011, pp. 2567-2571.
90. V. Rathi, R. Urbanke, M. Andersson, and M. Skoglund, " Rateequivocation optimal spatially coupled LDPC codes for the BEC wiretap channel," in Proc. IEEE Int. Symp. Inf. Theory, 5, 2011, pp. 2393-2397.
91. H. Uchikawa, B. Kurkoski, K. Kasai, and K. Sakaniwa, "Threshold improvement of low-density lattice codes via spatial coupling," in Proc. Int. Conf. Comput., Netw. Commun., 2, 2012, pp. 1036-1040.
92. A. Yedla, P. Nguyen, H. Pfister, and K. Narayanan, "Universal codes for the Gaussian MAC via spatial coupling," in Proc. 49th Annu. Allerton Conf. Commun., Control, Comput., Sep. 2011, pp. 1801-1808.
93. P.Nguyen, A. Yedla,H. Pfister, and K. Narayanan, "Threshold saturation of spatially-coupled codes on intersymbol-interference channels," in Proc. IEEE Int. Conf. Commun., Jun. 2012, pp. 2181-2186.
94. S. H. Hassani, N. Macris, and R. Urbanke, Coupled graphical models and their thresholds, presented at the IEEE Inf. Theory Workshop, Dublin, Ireland, Sep. 2010.
95. S. H. Hassani, N. Macris, and R. Urbanke, Thresholds saturation in spatially coupled constraint satisfaction problems, Dec. 2011 [Online]. Available: http://arxiv.org/abs/1112.6320
96. S. H. Hassani, N. Macris, and R. Urbanke, "Chains of mean-field models," J. Statist. Mech.: Theory Exp., vol. 2012, no. 02, p. P02011, 2012 [Online]. Available: http://stacks.iop.org/1742-5468/2012/i=02/ a=P02011
97. K. Takeuchi, T. Tanaka, and T. Kawabata, "A phenomenological study on threshold improvement via spatial coupling," IEICE Trans. Fund. Electron., Commun. Comput. Sci., vol. E95, pp. 974-977, 2012.
98. A. Giurgiu, N. Macris, and R. Urbanke, "How to prove the Maxwell conjecture via spatial coupling: A proof of concept," in Proc. IEEE Int. Symp. Inf. Theory, Jul. 2012, pp. 458-462.
99. B. Bollobás, Random Graphs. Cambridge, U.K.: Cambridge Univ. Press, 2001.
100. G. Hanoch and H. Levy, "The efficiency analysis of choices involving risk," Rev. Econ. Stud., vol. 36, pp. 335-346, 1969.
101. S. Huettinger and J. B. Huber, "Design of "multiple-turbo codes" with transfer characteristics of component codes," presented at the Conf. Inf. Sci. Syst., Princeton, NJ, USA, Mar. 2002.
102. S. Huettinger and J. B. Huber, "Information processing and combining in channel coding," in Proc. Int. Conf. Turbo Codes Relat. Top., Brest, France, Sep. 2003, pp. 95-102.
103. I. Land, P. Hoeher, S. Huettinger, and J. B. Huber, "Bounds on information combining," in Proc. Int. Conf. Turbo Codes Relat. Top., Brest, France, Sep. 2003, pp. 39-42.
104. I. Sutskover, S. Shamai, and J. Ziv, "Extremes of information combining," presented at the Allerton Conf. Commun., Control, Comput., Monticello, IL, USA, Oct. 2003.
105. I. Sutskover, S. Shamai, and J. Ziv, "Extremes of information combining," IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1313-1325, Apr. 2005. (Pubitemid 40555877)
106. C. Villani, Optimal Transport, Old and New. NewYork,NY,USA: Springer-Verlag, 2009, vol. 338.
107. V. M. Zolotarev, Modern Theory of Summation of Random Variables. Utrecht, The Netherlands: VSP, 1997.
108. S. t. Brink, "Designing iterative decoding schemes with the extrinsic information transfer chart," AEU Int. J. Electron. Commun., vol. 54, pp. 389-398, Dec. 2000.
109. A. Ashikhmin, G. Kramer, and S. T. Brink, "Extrinsic information transfer functions: Model and erasure channel property," IEEE Trans. Inf. Theory, vol. 50, no. 11, pp. 2657-2673, Nov. 2004.
110. C. Méasson, A. Montanari, and R. Urbanke, "Maxwell construction: The hidden bridge between maximum-likelihood and iterative decoding," IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5277-5307, Dec. 2008.
111. M. B. Tavares, K. S. Zigangirov, and G. P. Fettweis, "Tail-biting LDPC convolutional codes," in Proc. IEEE Int. Symp. Inf. Theory, Jun. 2007, pp. 2341-2345.
112. A. K. Khandekar, "Graph-based codes and iterative decoding," Ph.D. dissertation, California Institute of Technology, Pasadena, CA, USA, 2002.
113. D. Burshtein and G. Miller, "Expander graph arguments for message- passing algorithms," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 782-790, Feb. 2001. (Pubitemid 32318103)
114. M. Lentmaier, D. Truhachev, K. Zigangirov, and D. J. Costello, Jr., "An analysis of the block error probability performance of iterative decoding," IEEE Trans. Inf. Theory, vol. 51, no. 11, pp. 3834-3855, Nov. 2005. (Pubitemid 41622584)
115. A. Yedla, Y.-Y. Jian, P. S. Nguyen, and H. D. Pfister, "A simple proof of threshold saturation for coupled scalar recursions," CoRR vol. abs/ 1204.5703, 2012.
116. A. Yedla, Y.-Y. Jian, P. S. Nguyen, and H. D. Pfister, "A simple proof of threshold saturation for coupled vector recursions," CoRR vol. abs/ 1208.4080, 2012.
117. S. Kudekar, T. Richardson, and R. L. Urbanke, "Wave-like solutions of general one-dimensional spatially coupled systems," CoRR vol. abs/ 1208.5273, 2012.
118. G. Wiechman and I. Sason, "Parity-check density versus performance of binary linear block codes: New bounds and applications," IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 550-579, Feb. 2007. (Pubitemid 46358436)
119. S. S. Vallender, "Calculation of theWasserstein distance between probability distributions on the line," Theor. Probability Appl., vol. 18, pp. 784-786, 1973.
120. L. Boczkowski, "Extended extremes of information combining," CoRR vol. abs/1201.6046, 2012.
121. R. Cauty, "Solution du problème de point fixe de Schauder," Fundam. Math., no. 170, pp. 231-246, 2001. (Pubitemid 33571859)