Performance analysis of linear codes under maximum-likelihood decoding: A tutorial

Foundations and Trends in Communications and Information Theory

Volumn 3, Issue 1-2, 2006, Pages 1-222

  Sason, Igal ^a   Shamai, Shlomo ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; ERROR ANALYSIS; GRAPH THEORY; ITERATIVE METHODS; LINEAR EQUATIONS; MAXIMUM LIKELIHOOD ESTIMATION;

GALLAGER BOUNDING TECHNIQUES; OPTIMAL MAXIMUM-LIKELIHOOD (ML) DECODING; PERFORMANCE EVALUATION;

DECODING;

EID: 33746600795     PISSN: 15672190     EISSN: 15672328     Source Type: Journal    
DOI: 10.1561/0100000009     Document Type: Article

References (230)

1. A. Abbasfar, K. Yao, and D. Divsalar, "Maximum-likelihood decoding analysis of accumulate-repeat-accumulate codes," in Proceedings IEEE 2004 Global Telecommunications Conference (GLOBECOM 2004), (Dallas, Texas, USA), pp. 514-519, November 29-December 3, 2004.
2. E. Agrell, "Voronoi regions for binary linear block codes," IEEE Trans. on Information Theory, vol. 42, pp. 310-316, January 1996.
3. E. Agrell, "On the Voronoi neighbor ratio for binary linear block codes," IEEE Trans. on Information Theory, vol. 44, pp. 3064-3072, November 1998.
4. S. Aji, H. Jin, A. Khandekar, R. J. McEliece, and D. J. C. Mackay, "BSC thresholds for code ensembles based on 'typical pairs' decoding," in Codes, Systems and Graphical Models, pp. 195-210, 2001. Springer-Verlag Series IMA Volumes in Mathematics and its Applications (B. Marcus and J. Rosental, eds.).
5. R. Annavajjala, A. Chockalingam, and L. B. Milstein, "Performance analysis of coded communication systems on Nakagami fading channels with selection combining diversity," IEEE Trans. on Communications, vol. 52, pp. 1214-1220, July 2004.
6. A. Ashikmin and A. Barg, "Minimal vectors in linear codes," IEEE Trans. on Information Theory, vol. 44, pp. 2010-2017, September 1998.
7. M. Aydinlik and M. Salehi, "Performance bounds for unequal error protecting turbo codes," in Proceedings 2005 Conference on Information Sciences and Systems (CISS 2005), (Johns Hopkins University, Maltimore, MD, USA), pp. 90-96, March 16-18, 2005.
8. F. Babich, G. Montorsi, and F. Vatta, "Performance bounds of continuous and blockwise decoded turbo codes in Rician fading channel," Electronics Letters, vol. 34, pp. 1646-1648, August 1998.
9. F. Babich, G. Montorsi, and F. Vatta, "Improved union bounds on turbo codes performance," IEE Proceedings on Communications, vol. 147, pp. 337-344, December 2000.
10. F. Babich, G. Montorsi, and F. Vatta, "On the Viterbi and Viterbi's improved union bounds on turbo codes performance," in Proceedings Second International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 527-531, September 4-7, 2000.
11. F. Babich, "On the performance of efficient coding techniques over fading channels," IEEE Trans. on Wireless Communications, vol. 3, pp. 290-299, January 2004.
12. V. B. Balakirsky and A. J. H. Vinck, "Estimates of the decoding error probability for parallel channels with dependent noise," in Proceedings IEEE International Symposium on Information Theory and its Applications (ISITA 2004), (Parma, Italy), pp. 1568-1573, October 2004.
13. A. M. Barg and I. I. Dumer, "On computing the weight spectrum of cyclic codes," IEEE Trans. on Information Theory, vol. 38, pp. 1382-1386, July 1992.
14. A. Barg, "On the asymptotic accuracy of the union bound," in Proceedings Forty-Second Annual Allerton Conference on Communication, Control and Computing, (Urbana-Champaign, IL, USA), pp. 1352-1361, September 2004.
15. L. Bazzi, T. Richardson, and R. Urbanke, "Exact thresholds and optimal codes for the binary symmetric channel and Gallager's decoding algorithm A," IEEE Trans. on Information Theory, vol. 50, pp. 2010-2021, September 2004.
16. F. Behnamfar, F. Alajaji, and T. Linder, "Improved lower bounds for the error rate of linear block codes," in Proceedings 43rd Allerton Conference on Control, Computing and Communications, (Monticello, Illinois, USA), pp. 2227-2236, September 2005.
17. S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, "Analysis, design, and iterative decoding of double serially concatenated codes with interleaves," IEEE Journal on Selected Areas in Communications, vol. 16, pp. 231-244, February 1998.
18. S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, "Serial concatenation of interleaved codes: Performance analysis, design and iterative decoding," IEEE Trans. on Information Theory, vol. 44, pp. 909-926, May 1998.
19. S. Benedetto and G. Montorsi, "Unveiling turbo codes: some results on parallel concatenated coding schemes," IEEE Trans. on Information Theory, vol. 42, pp. 409-429, March 1996.
20. S. Benedetto and G. Montorsi, "Performance of continuous and blockwise decoded turbo codes," IEEE Communications Letters, vol. 1, pp. 77-79, May 1997.
21. A. Bennatan and D. Burshtein, "On the application of LDPC codes to arbitrary discrete-memoryless channels," IEEE Trans. on Information Theory, vol. 50, pp. 417-438, March 2004.
22. E. R. Berlekamp, "The technology of error correction codes," Proceedings of the IEEE, vol. 68, pp. 564-593, May 1980.
23. C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding," in Proceedings 1993 IEEE International Conference on Communications (ICC'93), (Geneva, Switzerland), pp. 1064-1070, May 1993.
24. C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. on Communications, vol. 44, pp. 1261-1271, October 1996.
25. E. Biglieri, G. Caire, and G. Taricco, "Expurgating the union bound to error probability: A generalization of the Verdu-Shields theorem," in Proceedings of the 1997 IEEE International Symposium on Information Theory, (Ulm, Germany), June 1997.
26. E. Biglieri, J. Proakis, and S. Shamai, "Fading channels: Information-theoretic and communications aspects," IEEE Trans. on Information Theory, vol. 44, pp. 2619-2692, October 1998.
27. H. Bouzekri and S. L. Miller, "An upper bound on turbo codes performance over quasi-static fading channels," IEEE Communications Letters, vol. 7, pp. 302-304, July 2003.
28. M. Breiling and J. Huber, "A method for determining the distance profile of turbo codes," in Proceedings 3rd ITC Conference on Source and Channel Coding, (Munich, Germany), pp. 219-224, January 2000.
29. C. Brutel and J. Boutros, "Serial concatenation of interleaved convolutional codes and M-ary continuous phase modulations," Annals of Telecommunications, vol. 54, pp. 235-242, March-April 1999.
30. A. G. Burr and G. P. White, "Comparison of iterative decoder performance with union bounds for short frame turbo codes," Annals of Telecommunications, vol. 54, pp. 201-207, March-April 1999.
31. A. G. Burr, "Bounds on coding gain versus decoding delay for spherical codes on the Gaussian channel," in Proceedings 1997 IEEE International Symposium on Information Theory (ISIT 1997), (Ulm, Germany), June-July 1997.
32. A. G. Burr, "Multilevel turbo-coded modulation: performance bounds," in Proceedings of the First International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 111-118, September 1997.
33. D. Burshtein, M. Krivelevich, S. Litsyn, and G. Miller, "Upper bounds on the rate of LDPC codes," IEEE Trans. on Information Theory, vol. 48, pp. 2437-2449, September 2002.
34. D. Burshtein and G. Miller, "Asymptotic enumeration methods for analyzing LDPC codes," IEEE Transactions on Information Theory, vol. 50, pp. 1115-1131, June 2004.
35. G. Caire and E. B. G. Taricco and, "Bit-interleaved coded modulation," IEEE Trans. on Information Theory, vol. 44, pp. 927-946, May 1998.
36. G. Caire, G. Taricco, and G. Batail, "Weight distribution and performance of the iterated product of single-parity check-codes," in Proceedings 1994 IEEE Global Telecommunications Conference (GLOBECOM '94), Communications Theory Mini-Conference, pp. 206-211, December 1994.
37. M. Cedervall and R. Johannesson, "A fast algorithm for computing the distance spectrum of convolutional codes," IEEE Trans. on Information Theory, vol. 35, pp. 1146-1159, November 1989.
38. P. Chaudhari and A. K. Khandani, "Using the Fourier transform to compute the weight distribution of a binary linear block code," IEEE Communications Letters, vol. 5, pp. 22-24, January 2001.
39. A. Cohen and N. Merhav, "Lower bounds on the error probability of block codes based on improvements on de Caen's inequality," IEEE Trans. on Information Theory, vol. 50, pp. 290-310, February 2004.
40. J. Craig, "A new, simple and exact result for calculating error probability for two-dimensional signal constellation," in Proceedings of 1991 Military Communications Conference (MILCOM '91), (Boston, MA, USA), pp. 25.5.1-25.5.5, November 1991.
41. I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems. NY: Academic Press, 1981.
42. A. Dembo personal communications, communicated by I. Sason and S. Shamai, 2000.
43. Y. Desaki, T. Fujiwara, and T. Kasami, "A method for computing the weight distribution of a block code by using its trellis diagram," IEICE Trans. Fundamentals, vol. E77-A, pp. 1230-1237, August 1994.
44. Y. Desaki, T. Fujiwara, and T. Kasami, "The weight distributions of extended binary primitive BCH codes of length 128," IEEE Trans, on Information Theory, vol. 43, pp. 1364-1371, July 1997.
45. D. de Caen, "A lower bound on the probability of a union," Discrete mathematics, vol. 169, pp. 217-220, 1997.
46. H. M. de Oliveira and G. Battail, "The random coded modulation: Performance and Euclidean distance spectrum evaluation," Annals of Telecommunications, vol. 47, pp. 107-124, March-April 1992.
47. D. Divsalar and E. Biglieri, "Upper bounds to error probabilities of coded systems beyond the cuttoff rate," IEEE Trans, on Communications, vol. 51, pp. 2011-2018, December 2003.
48. D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer function bounds on the performance of turbo codes," Jet Propulsion Laboratory (JPL), CA, USA, TDA Progress Report 42-122, August 1995. [Online] Available: http://tmo.jpl.nasa.gov/tmo/progressreport/42-122/122A.pdf.
49. D. Divsalar, H. Jin, and R. J. McEliece, "Coding theorems for 'turbo-like' codes," in Proceedings of the 36th Allerton Conference on Communication, Control, and Computing, (Monticello, Illinois), pp. 201-210, September 23-25 1998.
50. D. Divsalar and F. Pollara, "Serial and hybrid concatenated codes with applications," in Proceedings of International Symposium on Turbo Codes and Related Topics, pp. 80-87, September 1997.
51. D. Divsalar, "Ensemble weight enumerators for protograph LDPC codes," in Proceedings IEEE 2006 International Symposium on Information Theory (ISIT 2006), (Seattle, Washington, USA), July 9-14 2006.
52. D. Divsalar, "A simple tight bound on error probability of block codes with application to turbo codes," the Telecommunications and Mission Operations (TMO) Progress Report 42-139, JPL, pp. 1-35, November 15, 1999. [Online] Available: http://tmo.jpl.nasa.gov/tmo/progress.report/42-139/139L.pdf.
53. C. Di, A. Montanari, and R. Urbanke, "Weight distributions of LDPC code ensembles: combinatorics meets statistical physics," in Proceedings 2004 IEEE International Symposium on Information Theory (ISIT 2004), (Chicago, USA), p. 102, July 2004.
54. C. Di, T. Richardson, and R. Urbanke, "Weight distribution of low-density parity-check codes," accepted to IEEE Trans. on Information Theory, 2006.
55. C. Di, Asymptotic and Finite-Length Analysis of Low-Density Parity-Check Codes. Ph.D. dissertation, EPFL, Lausanne, Switzerland, September 2004.
56. S. Dolinar, D. Divsalar, and F. Pollara, "Code performance as a function of block size," Jet Propulsion Laboratory (JPL), TMO Progress Report 42-133, pp. 1-23, May 15, 1998. [Online] Available: http://tmo.jpl.nasa. gov/tmo/progress_report/42-133/133K.pdf.
57. S. Dolinar and D. Divsalar, "Weight distributions for turbo codes using random and nonrandom permutations," Jet Propulsion Laboratory (JPL), CA, USA, TDA Progress Report 42-122, August 1995. [Online] Available: http://tmo.jpl.nasa.gov/tmo/progress_report/42-122/122B.pdf.
58. T. M. Duman and E. M. Kurtas, "Performance bounds for high rate linear codes over partial response channels," IEEE Trans. on Information Theory, vol. 47, pp. 1201-1205, March 2001.
59. T. M. Duman and M. Salehi, "New performance bounds for turbo codes," IEEE Trans, on Communications, vol. 46, pp. 717-723, June 1998.
60. T. M. Duman and M. Salehi, "Performance bounds for turbo-coded modulation systems," IEEE Trans. on Communications, vol. 47, pp. 511-521, April 1999.
61. T. M. Duman, Turbo Codes and Turbo Coded Modulation Systems: Analysis and Performance Bounds. Ph.D. dissertation, Elect. Comput. Eng. Dep., Northeastern University, Boston, MA, USA, May 1998.
62. T. Duman and E. Kurtas, "Maximum likelihood decoding bounds for high rate turbo codes over Loretnzian channels," Electronics Letters, vol. 37, pp. 771-773, June 2001.
63. T. Duman and M. Salehi, "The union bound for turbo coded modulation systems over fading channels," IEEE Transactions on Communications, vol. 47, pp. 1495-1502
64. (See also the correction in IEEE Trans. on Communications, vol. 50, p. 1766, November 1999), October 1999.
65. P. M. Ebert, Error Bounds for Parallel Communication Channels. Ph.D. dissertation, MIT, August 1966.
66. M. El-Khamy and R. Garello, "On the weight enumerator and the maximum-likelihood performance of linear product codes," in submitted to IEEE Trans. on Information Theory, December 2005. [Online]. Available: http://arxiv.org/abs/cs.IT/0601095.
67. M. El-Khamy and R. J. McEliece, "Bounds on the average binary minimum distance and the maximum-likelihood performance of Reed Solomon codes," in Proceedings Forty-Second Annual Allerton Conference on Communication, Control and Computing, (Urbana-Champaign, IL, USA), pp. 290-299, September 29-October 1, 2004.
68. K. Engdahl and K. Zigangirov, "Tighter bounds on the error probability of fixed convolutional codes," IEEE Trans. on Information Theory, vol. 47, pp. 1625-1630, May 2001.
69. q," IEEE Trans. On Information Theory, vol. 51, pp. 1871-1879, May 2005.
70. T. Etzion, A. Trachtenberg, and A. Vardy, "Which codes have cycle-free Tanner graphs?," IEEE Trans. on Information Theory, vol. 45, pp. 2173-2181, September 1999.
71. A. G. Fabregas and G. Caire, "Coded modulation in the block-fading channel: Coding theorems and code construction," IEEE Trans. on Information Theory, vol. 52, pp. 91-114, January 2006.
72. R. M. Fano, Transmission of Information. Jointly published by the MIT Press and John Wiley & Sons, 1961.
73. S. L. Fogal, R. McEliece, and J. Thorpe, "Enumerators for protograph ensembles of LDPC codes," in Proceedings IEEE 2005 International Symposium on Information Theory (ISIT 2005), (Adelaide, Australia), pp. 2156-2160, September 4-9 2005.
74. G. D. Forney and G. Ungerboeck, "Modulation and coding for linear Gaussian channels," IEEE Trans. on Information Theory, vol. 44, pp. 2384-2415, October 1998.
75. M. P. C. Fossorier, S. Lin, and D. J. Costello, "On the weight distribution of terminated convolutional codes," IEEE Trans, on Information Theory, vol. 45, pp. 1646-1648, July 1999.
76. M. Fossorier, "Critical point for maximum-likelihood decoding of linear block codes," IEEE Communications Letters, vol. 9, pp. 817-819, September 2005.
77. M. Fozunbal, S. W. McLaughlin, and R. W. Schafer, "On performance limits of space-time codes: A sphere-packing bound approach," IEEE Trans. on Information Theory, vol. 49, pp. 2681-2687, October 2003.
78. S. Franz, M. Leone, A. Montanari, and F. R. Tersenghi, "Dynamic phase transition for decoding algorithms," Physical Review E, vol. 66, pp. 04612-01-046120-17, October 2002.
79. F. Gagnon and D. Haccoun, "Bounds on the error performance of coding for non independent Rician Fading channels," IEEE Trans. on Communications, vol. 40, pp. 351-360, February 1992.
80. J. Galambos and I. Simonelli, "Bonferroni-type inequalities with Applications," in Springer Series in Statistics, Probability and its Applications, (New-York), Springer-Verlag, 1996.
81. R. G. Gallager, "Low-density parity-check codes," IRE Trans. on Information Theory, vol. 8, pp. 21-28, January 1962.
82. R. G. Gallager, Low-density parity-check codes. Cambridge, MA, USA: MIT Press, 1963.
83. R. G. Gallager, "A simple derivation of the coding theorem and some applications," IEEE Trans. on Information Theory, vol. 11, pp. 3-18, January 1965.
84. R. G. Gallager, Information Theory and Reliable Communications. 1968.
85. R. G. Gallager, "The random coding bound is tight for the average code," IEEE Trans. on Information Theory, vol. 19, pp. 244-246, March 1973.
86. A. Ganti, A. Lapidoth, and E. Telatar, "Mismatched decoding revisited: General alphabets, channels with memory, and the wide-band limit," IEEE Trans. on Information Theory, vol. 46, pp. 2315-2328, November 2000.
87. I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series and products. Academic Press, Fifth ed., 1994.
88. S. Guemghar and G. Caire, "On the performance of finite length irregular repeat-accumulate codes," in Proceedings of the Fifth ITG Conference on Source and Channel Coding, (Germany), pp. 79-86, January 14-16, 2004.
89. E. K. Hall and S. G. Wilson, "Design and analysis of turbo codes on Rayleigh fading channels," IEEE Journal on Selected Areas in Communications, vol. 16, pp. 160-174, February 1998.
90. J. Ha, J. Kim, and S. W. McLaughlin, "Rate-compatible puncturing of low-density parity-check codes," IEEE Trans, on Information Theory, vol. 50, pp. 2824-2836, November 2004.
91. H. Herzberg and G. Poltyrev, "Techniques for bounding the probability of decoding error for block coded modulations structures," IEEE Trans. on Information Theory, vol. 40, pp. 903-911, May 1994.
92. H. Herzberg and G. Poltyrev, "The error probability of M-ary PSK block coded modulation schemes," IEEE Trans. on Communications, vol. 44, pp. 427-433, April 1996.
93. J. Hokfelt, O. Edfors, and T. Maseng, "On the theory and performance of trellis termination methods for turbo codes," IEEE Journal on Selected Areas in Communications, vol. 19, pp. 838-847, May 2001.
94. C. H. Hsu and A. Anastasopoulos, "Asymptotic weight distributions of irregular repeat-accumulate codes," in Proceedings 2005 IEEE Global Telecommunications Conference (GLOBECOM '05), (St Louis, Mo, USA), pp. 1147-1151, November 2005.
95. C. H. Hsu and A. Anastasopoulos, "Capacity-achieving codes with bounded graphical complexity on noisy channels," in presented in the 43rd Allerton Conference on Communication, Control and Computing, (Monticello, Illinois, USA), pp. 28-30, September 2005. [Online]. Available: http://www.arxiv.org/abs/cs.IT/0509062.
96. C. H. Hsu and A. Anastasopoulos, "Capacity-achieving LDPC codes through puncturing," in Proceedings 2005 IEEE International Conference on Wireless Networks and Mobile Computing (WirelessCom 2005), (Mauii, Hawaii, USA), pp. 13-16, June 2005.
97. X. L. Huang, N. Phamdo, and L. Ping, "BER bounds on parallel concatenated single parity-check arrays and Zigzag codes," in Proceedings 1999 IEEE Conference on Global Communications (GLOBECOM '99), (Rio de Janeiro, Brazil), pp. 2436-2440, December 1999.
98. B. Hughes, "On the error probability of signals in additive white Gaussian noise," IEEE Trans. on Information Theory, vol. 37, pp. 151-155, January 1991.
99. L. W. Hughes, "A simple upper bound on the error probability for orthogonal signals in white noise," IEEE Trans. on Communications, vol. 40, p. 670, April 1992.
100. D. Hunter, "An upper bound for the probability of a union," Journal of Applied Probability, vol. 13, pp. 597-603, 1976.
101. J. Hu and S. L. Miller, "An improved upper bound on the performance of convolutional codes over quasi-static fading channels," in Proceedings 2003 IEEE Global Communications Conference (GLOBECOM 2003), (San Francisco, CA, USA), pp. 1593-1597, December 1-5, 2003.
102. E. A. Ince, N. S. Kambo, and S. A. Ali, "Efficient expression and bound for pairwise error probability in Rayleigh fading channels, with application to union bounds for turbo codes," IEEE Communications Letters, vol. 9, pp. 25-27, January 2005.
103. J. Jiang and K. R. Narayanan, "Iterative soft input soft output decoding of Reed-Solomon codes by adapting the parity-check matrix," submitted to IEEE Trans. on Information Theory, [Online]. Available: http://www.arxiv.org/list/cs.IT/0506073, June 2005.
104. H. Jin and R. J. McEliece, "RA codes achieve AWGN channel capacity, lecture notes in computer science," in Proceedings in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 13th International Symposium (AAECC-13), (M. Fossorier, H. Imai, S. Lin, and A. Poli, eds.), (Honolulu, Hawaii, USA), pp. 10-18, Springer-Verlag Heidelberg, November 1999.
105. H. Jin and R. J. McEliece, "Coding theorems for turbo-like ensembles," IEEE Trans, on Information Theory, vol. 48, pp. 1451-1461, June 2002.
106. G. Kaplan and S. Shamai, "Information rates and error exponents of compound channels with application to antipodal signaling in a fading environment," International Journal of Electronics and Communication (AEU), vol. 47, pp. 228-239, April 1993.
107. G. Kaplan and S. Shamai, "Achievable performance over the correlated Rician channel," IEEE Trans. on Communications, vol. 42, pp. 2967-2978, November 1994.
108. T. Kasami, T. Fujiwara, and S. Lin, "Approximation of the weight distribution of binary linear block codes," IEEE Trans. on Information Theory, vol. 31, pp. 769-780, November 1985.
109. O. Keren and S. Litsyn, "A lower bound on the probability of decoding error over a BSC channel," in Proceedings 21st IEEE Convention of the Electrical and Electronic Engineers in Israel, (Tel-Aviv, Israel), pp. 271-273, April 2000.
110. R. Knopp and P. A. Humblet, "On coding for block fading channels," IEEE Trans, on Information Theory, vol. 46, pp. 189-205, January 2000.
111. K. Koiko and H. Ogiwara, "Performance evaluation of turbo codes over impulsive noise channels," IEICE Trans. Fundamentals, vol. E84-A, pp. 2418-2426, October 2001.
112. P. Komulainen and K. Pehkonen, "Performance evaluation of superorthognal turbo codes in AWGN and flat Rayleight fading channels," IEEE Journal on Selected Areas in Communications, vol. 16, pp. 196-205, February 1998.
113. T. Koumoto, "Web site on the weight distribution of BCH and Reed-Muller codes," [Online]. Available: http://www.infsys.cne.okayama-u. ac. jp/~koumoto/wd/index.html, August 2004.
114. H. Kuai, F. Alajaji, and G. Takahara, "A lower bound on the probability of a finite union of events," Journal of Discrete Mathematics, vol. 215, pp. 147-158, March 2000.
115. A. Lapidoth and S. Shamai, "Fading channels: How perfect need "perfect side-information" be?," IEEE Trans. on Information Theory, vol. 48, pp. 1118-1130, May 2002.
116. D. L. Lazic, T. Beth, and M. Calic, "How close are turbo codes to optimal codes?," in Proceedings of the International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 192-195, September 3-5, 1997.
117. K. Leeuwin, J. C. Belfiore, and K. Kaleh, "Chernoff bound of trellis-coded modulation over correlated fading channels," IEEE Trans. on Communications, vol. 42, pp. 2506-2511, August 1994.
118. C. Ling and M. Fu, personal communications. October 2005.
119. S. Litsyn and V. Shevelev, "On ensembles of low-density parity-check codes: Asymptotic distance distributions," IEEE Trans. on Information Theory, vol. 48, pp. 887-908, April 2002.
120. R. Liu, P. Spasojevic, and E. Soljanin, "Reliable channel regions for good binary codes transmitted over parallel channels," IEEE Trans. on Information Theory, vol. 52, pp. 1405-1424, April 2006.
121. J. Li, K. R. Narayanan, and C. N. Georghiades, "Generalized product-accumulate codes: Analysis and performance," in Proceedings 2001 IEEE Global Communications Conference (GLOBECOM 2001), (San Antonio, Texas), pp. 975-979, November 25-29, 2001.
122. J. Li, K. R. Narayanan, and C. N. Georghiades, "Product-accumulate codes: A class of codes with near-capacity performance and low decoding complexity," IEEE Trans. on Information Theory, vol. 50, pp. 31-46, January 2004.
123. Y. Li and J. Moon, "Performance analysis of bit-interleaved space-time coding for OFDM in block fading channels," in Proceedings 2004 IEEE Vehicular Technology Conference (VTC2004-Spring), (Milan, Italy), May 2004. [Online]. Available: http://www-cdslab.ece.umn.edu/library/papers/vtc04S.pdf.
124. H. Lu, P. V. Kumer, and E. Yang, "On the input-output weight enumerators of product accumulate codes," IEEE Communications Letters, vol. 8, pp. 520-522, August 2004.
125. D. J. C. Mackay and R. M. Neal, "Near Shannon limit performance of low-density parity-check codes," IEEE Electronic Letters, vol. 33, pp. 457-458, March 1997.
126. S. J. Macmullan and O. M. Collins, "A comparison of known codes, random codes and the best codes," IEEE Trans. on Information Theory, vol. 44, pp. 3009-3022, November 1998.
127. E. Malkamaki and H. Leib, "Evaluating the performance of convolutional codes over block fading channels," IEEE Trans. on Information Theory, vol. 45, pp. 1643-1646, July 1999.
128. R. J. McEliece, "How to compute weight enumerators for convolutional codes," in Communications and Coding, (M. Darnel and B. Honary, eds.), ch. 6, pp. 121-141, 1998.
129. C. Measson, A. Montanari, T. Richardson, and R. Urbanke, "Life above threshold: From list decoding to area theorem and MSE," in 2004 IEEE Information Theory Workshop, (San Antonio, TX, USA), October 24-29, 2004. [Online]. Available: http://www.arxiv.org/abs/cs.IT/0410028.
130. C. Measson, A. Montanari, and R. Urbanke, "Maxwell construction: The hidden bridge between iterative and maximum a posteriori decoding," in IEEE Trans, on Information Theory, June 2005. submitted to [Online]. Available: http://www.arxiv.org/abs/cs.IT/0506083.
131. C. Measson, A. Montanari, and R. Urbanke, "Why we can not surpass capacity: The matching condition," presented in the 43rd Allerton Conference on Communication, Control and Computing, pp. 28-30, [Online]. Available: http://www.arxiv.org/abs/cs.IT/0510045, September 2005.
132. A. Mehrabian and S. Yousefi, "Improved tangential sphere bound on the ML decoding error probability of linear block codes in AWGN interference," in Proceedings 2005 Conference on Information Sciences and Systems (CISS 2005), (Maltimore, MD, USA), pp. 176-181, Johns Hopkins University, March 16-18, 2005.
133. N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, "On information rates for mismatched decoders," IEEE Trans. on Information Theory, vol. 40, pp. 1953-1967, November 1994.
134. G. Miller and D. Burshtein, "Bounds on the maximum-likelihood decoding error probability of low-density parity-check codes," IEEE Trans. on Information Theory, vol. 47, pp. 2696-2710, November 2001.
135. A. Mohammadi and W. Zhuang, "Variance of the turbo code performance bound over the interleaves," IEEE Trans. on Information Theory, vol. 48, pp. 2078-2086, July 2002.
136. A. Montanari, "Turbo codes: The phase transition," The European Physical Journal B, vol. 18, pp. 121-136, November 2000.
137. A. Montanari, "Tight bounds for LDPC codes and LDGM codes under MAP decoding," IEEE Trans, on Information Theory, vol. 51, pp. 3221-3246, September 2005.
138. P. Moqvist and T. M. Aulin, "Serially concatenated continuous phase modulation with iterative decoding," IEEE Trans, on Communications, vol. 49, pp. 1901-1915, November 2001.
139. A. H. Mugaibel and M. A. Kousa, "Evaluation of transfer functions for punctured turbo codes," Electronics Letters, vol. 36, pp. 805-807, April 2000.
140. M. Namokel, "Error performance bounds of turbo codes employing nonuniform interleaves," in Proceedings 1999 IEEE International Conference on Personal Wireless Communications, (Jaipur, India), pp. 404-408, February 1999.
141. K. R. Narayanan and G. L. Stüber, "A serial concatenation approach to iterative demodulation and decoding," IEEE Trans. on Communications, vol. 47, pp. 956-961, July 1999.
142. K. R. Narayanan and G. L. Stüber, "Performance of trellis-coded CPM with iterative demodulation and decoding," IEEE Trans. on Communications, vol. 49, pp. 676-687, April 2001.
143. N. Nefedov, "Evaluation of transfer functions for punctured turbo codes," in Proceedings 2nd International Symposium on Turbo Codes & Related Topics, (Brest, France), pp. 419-422, September 4-7, 2000.
144. M. Oberg and P. H. Siegel, "Performance analysis of turbo-equalized partial response channels," IEEE Trans. on Communications, vol. 49, pp. 436-444, March 2001.
145. T. Ohutsuki and J. M. Kahn, "BER performance of turbo-coded PPM CDMA systems on optical fiber," IEEE Journal of Lightwave Technology, vol. 18, pp. 1776-1784, December 2000.
146. T. Ohutsuki and J. M. Kahn, "Transfer function bounds on performance of binary turbo coding followed by M-ary ortohognal signal mapping through interleaver," in Proceedings 2000 IEEE International Conference on Communications (ICC 2000), (New Orleans, LA, USA), pp. 623-627, June 18-22, 2000.
147. I. Onyszchuk, Finding the complete path and weight enumerators of convolutional codes. February 1990. TDA Progress Report 42-100, JPL.
148. M. Peleg, I. Sason, S. Shamai, and A. Elia, "On interleaved, differntially encoded convolutional codes," IEEE Trans. on Information Theory, vol. 45, pp. 2572-2581, November 1999.
149. H. D. Pfister, I. Sason, and R. Urbanke, "Capacity-achieving ensembles for the binary erasure channel with bounded complexity," IEEE Trans. on Information Theory, vol. 51, pp. 2352-2379, July 2005.
150. H. D. Pfister and P. H. Siegel, "The serial concatenation of rate-1 codes through uniform random interleaves," IEEE Trans. on Information Theory, vol. 49, pp. 1425-1438, June 2003.
151. H. D. Pfister, On the capacity of finite state channels and the analysis of convolutional accumulate-m codes. March 2003.
152. L. Ping, W. K. Leung, and K. Y. Wu, "Low-rate turbo-hadamard codes," IEEE Trans. on Information Theory, vol. 49, pp. 3213-3224, December 2003.
153. G. S. Poltyrev, "Random coding bounds for discrete memoryless channels," Problems on Information Transmission, vol. 18, pp. 9-21,
154. translated from Problemy Predachi Informatsii (PPI), vol. 18, 12-26, 1982, January-March 1982.
155. G. Poltyrev, "Bounds on the decoding error probability of binary linear codes via their spectra," IEEE Trans. on Information Theory, vol. 40, pp. 1284-1292, July 1994.
156. A. Ramesh, A. Chockalingam, and L. B. Milstein, "Bounds on the performance of turbo codes on Nakagami fading channels with diversity combining," in Proceedings 2001 IEEE Global Telecommunications Conference (GLOBECOM 2001), (San Antonio, Texas, USA), pp. 1199-1204, November 25-29, 2001.
157. C. T. Retter, "An average weight-distance enumerator for binary expansions of Reed-Solomon codes," IEEE Trans. on Information Theory, vol. 48, pp. 1195-1200, May 2002.
158. T. Richardson, M. Shokrollahi, and R. Urbanke, "Design of capacity-approaching irregular low-density parity-check codes," IEEE Trans. on Information Theory, vol. 47, pp. 619-637, February 2001.
159. T. Richardson and R. Urbanke, "The capacity of low-density parity-check codes under message-passing decoding," IEEE Trans. on Information Theory, vol. 47, pp. 599-618, February 2001.
160. T. Richardson and R. Urbanke, "On the distribution of low weight codewords for turbo codes," in Proceedings Forty-Second Annual Allerton Conference on Communication, Control and Computing, (Urbana-Champaign, IL, USA), pp. 492-501, September 29-October 1, 2004.
161. T. Richardson and R. Urbanke, "Modern coding theory," in preparation. [Online]. Available: http://lthcwww.epfl.ch/papers/ics.ps.
162. E. Rosnes and O. Ytrehus, "An efficient algorithm for tailbiting turbo code weight distribution calculation," in Proceedings Third International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 439-442, September 1-5, 2003.
163. E. Rosnes and O. Ytrehus, "Improved algorithms for the determination of turbo code weight distributions," IEEE Trans. on Communications, vol. 53, pp. 20-26, January 2005.
164. M. Rouanne and D. J. Costello, "An algorithm for computing the distance spectrum of trellis codes," IEEE Journal on Selected Areas in Comunications, vol. 7, pp. 929-940, August 1989.
165. M. M. Salah, R. A. Raines, M. A. Temple, and T. G. Bailey, "Approach for deriving performance bounds of punctured turbo codes," Electronics Letters, vol. 35, pp. 2191-2192, December 1999.
166. I. Sason and I. Goldenberg, "Coding for parallel channels: Gallager bounds for binary linear codes with applications to turbo-like codes," submitted to the IEEE Trans. on Onformation Theory, June 2006.
167. I. Sason and I. Goldenberg, "On coding for independent and memoryless parallel channels: Tightened performance bounds and achievable channel regions for linear codes," submitted to the 24th IEEE Convention of Electrical and Electronics Engineers in Israel, November 15-17, 2006.
168. I. Sason, S. Shamai, and D. Divsalar, "Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels," IEEE Trans. on Communications, vol. 51, pp. 1296-1305, August 2003.
169. I. Sason and S. Shamai, "Gallager's 1961 bound: Extensions and observations," Technical Report, CC No. 258, Technion, Israel, October 1998.
170. I. Sason and S. Shamai, "Bounds on the error probability for block and turbo-block codes," Annals of Telecommunications, vol. 54, pp. 183-200, March-April 1999.
171. I. Sason and S. Shamai, "Improved upper bounds on the ensemble performance of ML decoded low-density parity-check codes," IEEE
172. I. Sason and S. Shamai, "Improved upper bounds on the ML decoding error probability of parallel and serial concatenated turbo codes via their ensemble distance spectrum," IEEE Trans. on Information Theory, vol. 46, pp. 24-47, January 2000.
173. I. Sason and S. Shamai, "On union bounds for random serially concatenated turbo codes with maximum likelihood decoding," European Trans. on Telecommunications, vol. 11, pp. 271-282, May-June 2000.
174. I. Sason and S. Shamai, "On improved bounds on the decoding error probability of block codes over interleaved fading channels, with applications to turbo-like codes," IEEE Trans. on Information Theory, vol. 47, pp. 2275-2299, September 2001.
175. I. Sason, E. Telatar, and R. Urbanke, "On the asymptotic input-output weight distributions and thresholds of convolutional and turbo-like encoders," IEEE Trans. on Information Theory, vol. 48, pp. 3052-3061, December 2002.
176. I. Sason and R. Urbanke, "Information-theoretic lower bounds on the bit error probability of codes on graphs," in Proceedings 2003 IEEE International Symposium on Information Theory, (Yokohama, Japan), p. 268, June 29-July 4, 2003.
177. I. Sason and R. Urbanke, "Parity-check density versus performance of binary linear block codes over memoryless symmetric channels," IEEE Trans. on Information Theory, vol. 49, pp. 1611-1635, July 2003.
178. I. Sason and G. Wiechman, "On achievable rates and complexity of LDPC codes for parallel channels with application to puncturing," submitted to IEEE Trans. on Information Theory, See http://arxiv.org/abs/cs.IT/0508072, August 2005.
179. I. Sason and G. Wiechman, "On achievable rates and complexity of LDPC codes for parallel channels with application to puncturing," in Proceedings IEEE 2006 IEEE International Symposium on Information Theory (ISIT 2006), (Seatle, Washington, USA), July 9-14, 2006.
180. I. Sason and G. Wiechman, "Performance versus complexity per iteration for low-density parity-check codes: An information-theoretic approach," in Proceedings of the Fourth International Symposium on Turbo Codes and Related Topics, (Munich, Germany), April 3-7, 2006.
181. I. Sason, Upper bounds on the maximum-likelihood decoding error probability for block codes and turbo-like codes. Ph.D. dissertation, Technion-Israel Institute of Technology, Haifa, Israel, September 2001. [Online]. Available: http://www.ee.technion.ac.il/people/sason/phd.html.
182. C. Schlegel and L. Perez, "On error bounds and turbo codes," IEEE Communications Letters, vol. 3, pp. 205-207, July 1999.
183. G. E. Seguin, "A lower bound on the error probability for signals in white Gaussian noise," IEEE Trans. on Information Theory, vol. 44, pp. 3168-3175, November 1998.
184. S. Shamai and I. Sason, "Variations on the gallager bounds, connections and applications," IEEE Trans. on Information Theory, vol. 48, pp. 3029-3051, December 2002.
185. C. E. Shannon, "Probability of error for optimal codes in a Gaussian channel," Bell System Technical Journal, vol. 38, pp. 611-656, May 1959.
186. C. Shannon, R. Gallager, and E. Berlekamp, "Lower bounds to error probability for coding on discrete memoryless channels," Parts I and II, Information and Control, vol. 10, pp. 65-103 and 522-552, February/May 1967.
187. H. Shin and J. H. Lee, "Improved upper bound on the bit error probability of turbo codes for ML decoding with perfect CSI in a Rayleigh fading channel," in Proceedings 2001 IEEE 12th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2001), (San Diego, CA, USA), pp. A.169-A.173, September 2001.
188. N. Shulman and M. Feder, "Random coding techniques for nonrandom codes," IEEE Trans. on Information Theory, vol. 45, pp. 2101-2104, September 1999.
189. M. K. Simon and D. Divsalar, "Some new twists to problems involving the Gaussian probability integral," IEEE Trans. on Communications, vol. 46, pp. 200-210, February 1998.
190. N. S. Skantzos, J. van Mourik, D. Saad, and Y. Kabashima, "Average and reliability error exponents in low-density parity-check codes," Journal of Physics A, vol. 36, pp. 11131-11141, October 2003.
191. T. Sugita, T. Kasami, and T. Fujiwara, "The weight distribution of the third-order Reed-muller code of length 512," IEEE Trans. on Information Theory, vol. 42, pp. 1622-1625, September 1996.
192. Y. V. Svirid, "Weight distributions and bounds for turbo codes," European Transactions on Telecommunications, vol. 6, pp. 543-555, September-October 1995.
193. P. F. Swaszek, "A lower bound on the error probability for signals in white Gaussian noise," IEEE Trans. on Information Theory, vol. 41, pp. 837-841, May 1995.
194. E. Telatar and R. Urbanke, "On the ensemble performance of turbo codes," in Proceedings 1997 IEEE International Symposium on Information Theory (ISIT 97), (Ulm, Germany), p. 105, June 1997.
195. C. Tellambura and A. D. S. Jayalath, "Generation of bivariate Rayleigh and Nakagami-m fading envelopes," IEEE Communications Letters, vol. 4, pp. 170-172, May 2000.
196. L. Tolhuizen, C. Baggen, and E. H. Nowacka, "Union bounds on the performance of product codes," in Proceedings 1998 IEEE International Symposium on Information Theory (ISIT 1998), (MIT, Cambridge, MA, USA), p. 267, August 15-21, 1998.
197. L. Tolhuizen and C. Baggen, "On the weight enumerator of product codes," Discrete Mathematics, vol. 106-107, 483-488, 1992.
198. L. Tolhuizen, "More results on the weight enumerator of product codes," IEEE Trans. on Information Theory, vol. 48, pp. 2573-2577, September 2002.
199. H. M. Tullberg and P. H. Siegel, "Serial concatenated TCM with inner accumulate code - Part 1: Maximum-likelihood analysis," IEEE Trans. on Communications, vol. 53, pp. 64-73, January 2005.
200. M. Twitto, I. Sason, and S. Shamai, "Tightened upper bounds on the ML decoding error probability of binary linear block codes," submitted to the IEEE Trans. on Information Theory, [Online]. Available: http://www.ee. technion.ac.il/people/sason/SFB.pdf, February 2006.
201. M. Twitto, I. Sason, and S. Shamai, "Tightened upper bounds on the ML decoding error probability of binary linear codes," accepted to 2006 IEEE International Symposium on Information Theory (ISIT 2006) Seatle, pp. 9-14, July 2006.
202. M. Twitto and I. Sason, "On the error exponents of some versions of improved tangential-sphere bounds," submitted to the IEEE Trans. on Information Theory, [Online]. Available: http://www.ee.technion.ac.il/people/ sason/error_exponents_itsb.pdf, March 2006.
203. R. Urbanke, "LDPC asymptotic weight and stopping set spectrum calculator," [Online]. Available: http://lthcwww.epfl.ch/~cdi/ldpc/ ldpc_define.php.
204. A. Valembois and M. Fossorier, "Box and match techniques applied to soft-decision decoding," IEEE Trans. on Information Theory, vol. 50, pp. 796-810, May 2004.
205. A. Valembois and M. Fossorier, "Sphere-packing bounds revisited for moderate block length," IEEE Trans. on Information Theory, vol. 50, pp. 2998-3014, December 2004.
206. S. Verdú, Multiuser detection. Cambridge University Press, 1998.
207. A. J. Viterbi and J. Omura, Principles of digital communications and coding. 1979.
208. A. J. Viterbi, A. M. Viterbi, J. Nicolas, and N. T. Sindushayana, "Perspectives on interleaved concatenated codes with iterative soft-output decoding," in Proceedings First International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 47-54, September 3-5 1997.
209. A. J. Viterbi, ch. 8, pp. 242-244, Principles of coherent communication. McGraw-Hill, 1966.
210. A. M. Viterbi and A. J. Viterbi, "An improved union bound for binary linear codes on the AWGN channel, with application to turbo decoding," in Proceedings of IEEE Information Theory Workshop, (San Diego, California, USA), p. 72, February 1998.
211. A. M. Viterbi and A. J. Viterbi, "Improved union bound on linear codes for the binary-input AWGN channel, with application to turbo codes," in Proceedings 1998 IEEE International Symposium on Information Theory (ISIT 1998), (MIT, Cambridge, MA, USA), p. 29, August 16-21, 1998.
212. A. M. Viterbi and A. J. Viterbi, "New results on serial concatenated and accumulated convolutional turbo code performance," Annals of Telecommunications, vol. 54, pp. 173-182, March-April 1999.
213. U. Wachsmann, R. F. H. Fischer, and J. B. Huber, "Multilevel codes: Theoretical concepts and practical design rules," IEEE Trans. on Information Theory, vol. 45, pp. 1361-1391, July 1999.
214. T. Weijun, R. M. Todd, and J. R. Cruz, "Bounds for low-density parity-check codes over partial response channels," IEEE Trans. on Magnetics, vol. 38, pp. 2310-2312, September 2002.
215. G. Wiechman and I. Sason, "Improved bounds on the parity-check density and achievable rates of binary linear block codes with applications to LDPC codes," submitted to IEEE Trans. on Information Theory, See http://arxiv.org/abs/cs.IT/0505057, May 2005.
216. G. Wiechman and I. Sason, "Improved bounds on the parity-check density and achievable rates of LDPC codes," in Proceedings Forty-Third Annual Allerton Conference on Communications, Control and Computing, pp. 1747-1758, September 28-30, 2005. See http://arxiv.org/abs/cs.IT/0505078.
217. G. Wiechman and I. Sason, "Improved sphere-packing bounds for codes of short to moderate block lengths with application to turbo-like codes," to be presented in the Forty-Four Annual Allerton Conference on Communications, Control and Computing, September 27-29, 2006.
218. J. K. Wolf and A. J. Viterbi, "On the weight distribution of linear block codes formed from convolutional codes," IEEE Trans. on Communications, vol. 44, pp. 1049-1051, September 1996.
219. K. Wu, L. Ping, X. Huang, and N. Phamdo, "Performance analysis of turbo-SPC codes," IEEE Trans. on Information Theory, vol. 50, pp. 2490-2494, October 2004.
220. X. Wu, Y. Cheng, and H. Xiang, "The Engdahl-Zigangirov bound for binary coded systems over block fading channels," IEEE Communications Letters, vol. 9, pp. 726-728, August 2005.
221. X. Wu and H. Xiang, "New gallager bounds in block fading channels," accepted to IEEE Trans. on Information Theory, June 2005.
222. K. Yasunaga and T. Fujiwara, "Relations between the local weight distributions of a linear block code, its extended code, and its even weight subcode," in Proceedings 2005 IEEE International Symposium on Information Theory (ISIT 2005), (Adelaide, Australia), pp. 382-386, September 2005.
223. T. Yokokawa, T. Miyauchi, K. Yamamoto, Y. Iida, M. Hattori, and R. J. McEliece, "ML Performance analysis method for SCCC and SCTCM with an 'in-line' interleaver," in Proceedings Third International Symposium on Turbo Codes and Related Topics, (Brest, France), pp. 303-306, September 1-5, 2003.
224. H. Yoshikawa, "On the calculation method of input-output weight distribution of terminated convolutional codes," in Proceedings 2003 International Symposium on Information Theory and Applications (ISITA 2004), (Parma, Italy), pp. 852-855, October 10-13, 2004.
225. S. Yousefi and A. K. Khandani, "Generalized tangential-sphere bound on the ML decoding error probability of linear binary block codes in AWGN interference," IEEE Trans. on Information Theory, vol. 50, pp. 2810-2815, November 2004.
226. S. Yousefi and A. K. Khandani, "A new upper bound on the ML decoding error probability of linear block codes in the AWGN channel," IEEE Trans. on Information Theory, vol. 50, pp. 3026-3036, December 2004.
227. J. Zangl and R. Herzog, "Improved tangential sphere bound on the bit error probability of concatenated codes," IEEE Journal on Selected Areas in Communications, vol. 19, pp. 825-830, May 2001.
228. J. Zheng and S. L. Miller, "Performance analysis of coded OFDM systems over frequency-selective fading channels," in Proceedings 2003 IEEE Global Telecommunications Conference (GLOBECOM '03), (San Francisco, CA, USA), pp. 1623-1627, December 1-5, 2003.
229. S. A. Zummo and W. E. Stark, "Performance analysis of coded systems over block fading channels," in Proceedings 2002 IEEE Vehicular Technology Conference (VTC 2002-Fall), pp. 1129-1133, September 24-28, 2002.
230. S. A. Zummo and W. E. Stark, "Performance analysis of binary coded systems over rician block fading channels," in Proceedings 2003 IEEE Military Communications Conference (MILCOM 2003), pp. 314-319, October 13-16, 2003.