Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial

This article is focused on the performance evaluation of linear codes under optimal maximum-likelihood (ML) decoding. Though the ML decoding algorithm is prohibitively complex for most practical codes, their performance analysis under ML decoding allows to predict their performance without resorting to computer simulations. It also provides a benchmark for testing the sub-optimality of iterative (or other practical) decoding algorithms. This analysis also establishes the goodness of linear codes (or ensembles), determined by the gap between their achievable rates under optimal ML decoding and information theoretical limits. In this article, upper and lower bounds on the error probability of linear codes under ML decoding are surveyed and applied to codes and ensembles of codes on graphs. For upper bounds, we discuss various bounds where focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, and also consider sphere-packing bounds with their recent improvements targeting codes of moderate block lengths.

[1]  Osnat Keren,et al.  A lower bound on the probability of decoding error over a BSC channel , 2000, 21st IEEE Convention of the Electrical and Electronic Engineers in Israel. Proceedings (Cat. No.00EX377).

[2]  Tor Aulin,et al.  Serially concatenated continuous phase modulation with iterative decoding , 2001, IEEE Trans. Commun..

[3]  A. M. Viterbi,et al.  Improved union bound on linear codes for the input-binary AWGN channel, with applications to turbo codes , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[4]  F. Pollara,et al.  Serial concatenation of interleaved codes: performance analysis, design and iterative decoding , 1996, Proceedings of IEEE International Symposium on Information Theory.

[5]  Meir Feder,et al.  Random coding techniques for nonrandom codes , 1999, IEEE Trans. Inf. Theory.

[6]  G. Kaplan,et al.  On Information Rates for Mismatched Decoders , 1993, Proceedings. IEEE International Symposium on Information Theory.

[7]  Charles T. Retter An average weight-distance enumerator for binary expansions of Reed-Solomon codes , 2002, IEEE Trans. Inf. Theory.

[8]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[9]  Idan Goldenberg,et al.  Coding for Parallel Channels: Gallager Bounds and Applications to Repeat-Accumulate Codes , 2006, 2006 IEEE 24th Convention of Electrical & Electronics Engineers in Israel.

[10]  Rüdiger L. Urbanke,et al.  Capacity-achieving ensembles for the binary erasure channel with bounded complexity , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[11]  J. van Mourik,et al.  Average and reliability error exponents in low-density parity-check codes , 2003 .

[12]  Gérald E. Séguin A Lower Bound on the Error Probability for Signals in White Gaussian Noise , 1998, IEEE Trans. Inf. Theory.

[13]  Robert J. McEliece,et al.  BSC Thresholds for Code Ensembles Based on “Typical Pairs” Decoding , 2001 .

[14]  Robert G. Gallager,et al.  The random coding bound is tight for the average code (Corresp.) , 1973, IEEE Trans. Inf. Theory.

[15]  Igal Sason,et al.  On the Error Exponents of Improved Tangential Sphere Bounds , 2007, IEEE Transactions on Information Theory.

[16]  Tadao Kasami,et al.  A Method for Computing the Weight Distribution of a Block Code by Using Its Trellis Diagram (Special Section on Information Theory and Its Applications) , 1994 .

[17]  Haige Xiang,et al.  The Engdahl-Zigangirov bound for binary coded systems over block-fading channels , 2005, IEEE Commun. Lett..

[18]  Xiaoling Huang,et al.  BER bounds on parallel concatenated single parity check arrays and zigzag codes , 1999, Seamless Interconnection for Universal Services. Global Telecommunications Conference. GLOBECOM'99. (Cat. No.99CH37042).

[19]  Elwyn R. Berlekamp,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II , 1967, Inf. Control..

[20]  Shlomo Shamai,et al.  On improved bounds on the decoding error probability of block codes over interleaved fading channels, with applications to turbo-like codes , 2001, IEEE Trans. Inf. Theory.

[21]  T. Richardson,et al.  On the Distribution of Low-Weight Codewords for Turbo Codes , 2004 .

[22]  Rüdiger L. Urbanke,et al.  Parity-check density versus performance of binary linear block codes over memoryless symmetric channels , 2003, IEEE Transactions on Information Theory.

[23]  Ove Edfors,et al.  On the theory and performance of trellis termination methods for turbo codes , 2001, IEEE J. Sel. Areas Commun..

[24]  Emre Telatar,et al.  Mismatched decoding revisited: General alphabets, channels with memory, and the wide-band limit , 2000, IEEE Trans. Inf. Theory.

[25]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[26]  Achilleas Anastasopoulos,et al.  Asymptotic weight distributions of irregular repeat-accumulate codes , 2005, GLOBECOM '05. IEEE Global Telecommunications Conference, 2005..

[27]  S. Franz,et al.  Dynamic phase transition for decoding algorithms. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  T. Etzion,et al.  Which codes have cycle-free Tanner graphs? , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[29]  Paul H. Siegel,et al.  On the capacity of finite state channels and the analysis of convolutional accumulate-m codes , 2003 .

[30]  Syed Amjad Ali,et al.  Efficient expression and bound for pairwise error probability in Rayleigh fading channels, with application to union bounds for turbo codes , 2005, IEEE Communications Letters.

[31]  Amir K. Khandani,et al.  Using the Fourier transform to compute the weight distribution of a binary linear block code , 2001, IEEE Communications Letters.

[32]  Shlomo Shamai,et al.  On Union Bounds for Random Serially Concatenated Turbo Codes with Maximum Likelihood Decoding , 2000, Eur. Trans. Telecommun..

[33]  Dariush Divsalar,et al.  Coding theorems for 'turbo-like' codes , 1998 .

[34]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[35]  T. Ohtsuki,et al.  BER performance of turbo-coded PPM CDMA systems on optical fiber , 2000, Journal of Lightwave Technology.

[36]  Andrea Montanari,et al.  Why We Can Not Surpass Capacity: The Matching Condition , 2005, ArXiv.

[37]  Fulvio Babich,et al.  Improved union bounds on turbo codes performance , 2000 .

[38]  David Haccoun,et al.  Bounds on the error performance of coding for nonindependent Rician-fading channels , 1992, IEEE Trans. Commun..

[39]  Shlomo Shamai,et al.  On interleaved, differentially encoded convolutional codes , 1999, IEEE Trans. Inf. Theory.

[40]  Masoud Salehi,et al.  Turbo codes and turbo coded modulation systems: analysis and performance bounds , 1998 .

[41]  Giorgio Taricco,et al.  Expurgating the union bound to error probability: a generalization of the Verdu-Shields theorem , 1997, Proceedings of IEEE International Symposium on Information Theory.

[42]  Simon Litsyn,et al.  On ensembles of low-density parity-check codes: Asymptotic distance distributions , 2002, IEEE Trans. Inf. Theory.

[43]  Maan A. Kousa,et al.  Evaluation of transfer functions for punctured turbo codes , 2000 .

[44]  Dariush Divsalar,et al.  Some new twists to problems involving the Gaussian probability integral , 1998, IEEE Trans. Commun..

[45]  Igal Sason,et al.  Bornes de la probabilité d’erreur pour des codes en blocs et des turbo codes en blocs avec décodage à maximum de vraisemblance , 1999 .

[46]  Laurence B. Milstein,et al.  Performance analysis of coded communication systems on Nakagami fading channels with selection combining diversity , 2004, IEEE Transactions on Communications.

[47]  H. Herzberg,et al.  Techniques of bounding the probability of decoding error for block coded modulation structures , 1994, IEEE Trans. Inf. Theory.

[48]  I. Sason,et al.  Log-Domain Calculation of the 1959 Sphere-Packing Bound with Application to M-ary PSK Block Coded Modulation , 2006, 2006 IEEE 24th Convention of Electrical & Electronics Engineers in Israel.

[49]  Scott L. Miller,et al.  An improved upper bound on the performance of convolutional codes over quasistatic fading channels , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[50]  Gregory Poltyrev,et al.  Bounds on the decoding error probability of binary linear codes via their spectra , 1994, IEEE Trans. Inf. Theory.

[51]  Roberto Garello,et al.  On the Weight Enumerator and the Maximum Likelihood Performance of Linear Product Codes , 2006, ArXiv.

[52]  Haruo Ogiwara,et al.  Performance Evaluation of Turbo Code over Impulsive Noise Channel , 2001 .

[53]  Jaekyun Moon,et al.  Performance analysis of bit-interleaved space-time coding for OFDM in block fading channels , 2004, 2004 IEEE 59th Vehicular Technology Conference. VTC 2004-Spring (IEEE Cat. No.04CH37514).

[54]  Mill Johannes G.A. Van,et al.  Transmission Of Information , 1961 .

[55]  R. Gallager Information Theory and Reliable Communication , 1968 .

[56]  I. Finding the Complete Path and Weight Enumerators of Convolutional Codes , .

[57]  Alexander Barg,et al.  On computing the weight spectrum of cyclic codes , 1992, IEEE Trans. Inf. Theory.

[58]  Andrew J. Viterbi,et al.  New results on serial concatenated and accumulated-convolutional turbo code performance , 1999, Ann. des Télécommunications.

[59]  Changyan Di Asymptotic and finite-length analysis of low-density parity-check codes , 2004 .

[60]  Fady Alajaji,et al.  A lower bound on the probability of a finite union of events , 2000, Discret. Math..

[61]  Robert J. McEliece,et al.  Coding theorems for turbo code ensembles , 2002, IEEE Trans. Inf. Theory.

[62]  Marc P. C. Fossorier Critical point for maximum likelihood decoding of linear block codes , 2005, IEEE Communications Letters.

[63]  Masoud Salehi,et al.  The union bound for turbo-coded modulation systems over fading channels , 1999, IEEE Trans. Commun..

[64]  T. Fujiwara,et al.  The weight distribution of the third-order Reed-Muller code of length 512 , 1996, IEEE Trans. Inf. Theory.

[65]  D. Divsalar A Simple Tight Bound on Error Probability of Block Codes with Application to Turbo Codes , 1999 .

[66]  Igal Sason,et al.  Improved Bounds on the Parity-Check Density and Achievable Rates of Binary Linear Block Codes with Applications to LDPC Codes , 2005, ArXiv.

[67]  Fulvio Babich,et al.  Performance bounds of continuous and blockwise decoded turbo codes in Rician fading channel , 1998 .

[68]  David J. Hunter An upper bound for the probability of a union , 1976, Journal of Applied Probability.

[69]  Jun Zheng,et al.  Performance analysis of coded OFDM systems over frequency-selective fading channels , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[70]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[71]  Jack K. Wolf,et al.  On the weight distribution of linear block codes formed from convolutional codes , 1996, IEEE Trans. Commun..

[72]  Krishna R. Narayanan,et al.  Iterative Soft-Input Soft-Output Decoding of Reed–Solomon Codes by Adapting the Parity-Check Matrix , 2005, IEEE Transactions on Information Theory.

[73]  Rüdiger L. Urbanke,et al.  Weight Distribution of Low-Density Parity-Check Codes , 2006, IEEE Transactions on Information Theory.

[74]  Jing Li,et al.  Product accumulate codes: a class of codes with near-capacity performance and low decoding complexity , 2004, IEEE Transactions on Information Theory.

[75]  Wayne E. Stark,et al.  Performance analysis of binary coded systems over Rician block fading channels , 2003, IEEE Military Communications Conference, 2003. MILCOM 2003..

[76]  Amiel Feinstein,et al.  Transmission of Information. , 1962 .

[77]  Achilleas Anastasopoulos,et al.  Capacity-Achieving Codes with Bounded Graphical Complexity on Noisy Channels , 2005, ArXiv.

[78]  Alain Glavieux,et al.  Reflections on the Prize Paper : "Near optimum error-correcting coding and decoding: turbo codes" , 1998 .

[79]  Laurence B. Milstein,et al.  Bounds on the performance of turbo codes on Nakagami fading channels with diversity combining , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[80]  Shu Lin,et al.  An approximation to the weight distribution of binary linear codes , 1985, IEEE Trans. Inf. Theory.

[81]  Igal Sason,et al.  Performance versus Complexity Per Iteration for Low-Density Parity-Check Codes: An Information-Theoretic Approach , 2005, ArXiv.

[82]  Achilleas Anastasopoulos,et al.  Capacity achieving LDPC codes through puncturing , 2005, 2005 International Conference on Wireless Networks, Communications and Mobile Computing.

[83]  Harry Leib,et al.  Evaluating the performance of convolutional codes over block fading channels , 1999, IEEE Trans. Inf. Theory.

[84]  Hideki Yoshikawa On the Calculation Method of Input-Output Weight Distribution of Terminated Convolutional Codes , 2004 .

[85]  David J. Goodman,et al.  Personal Communications , 1994, Mobile Communications.

[86]  G.L. Stuber,et al.  Performance of trellis coded CPM with iterative demodulation and decoding , 1999, Seamless Interconnection for Universal Services. Global Telecommunications Conference. GLOBECOM'99. (Cat. No.99CH37042).

[87]  David Burshtein,et al.  Bounds on the maximum-likelihood decoding error probability of low-density parity-check codes , 2000, IEEE Trans. Inf. Theory.

[88]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[89]  Rolf Johannesson,et al.  A fast algorithm for computing distance spectrum of convolutional codes , 1989, IEEE Trans. Inf. Theory.

[90]  A. Valembois,et al.  Box and match techniques applied to soft-decision decoding , 2002, Proceedings IEEE International Symposium on Information Theory,.

[91]  Wayne E. Stark,et al.  Performance analysis of coded systems over block fading channels , 2002, Proceedings IEEE 56th Vehicular Technology Conference.

[92]  G. Taricco,et al.  Weight distribution and performance of the iterated product of single-parity-check codes , 1994, 1994 IEEE GLOBECOM. Communications: Communications Theory Mini-Conference Record,.

[93]  Alister G. Burr,et al.  Comparison of iterative decoder performance with union bounds for short frame turbo codes , 1999, Ann. des Télécommunications.

[94]  Igal Sason,et al.  On Achievable Rates and Complexity of LDPC Codes for Parallel Channels with Application to Puncturing , 2005, ArXiv.

[95]  Tadao Kasami,et al.  The weight distributions of extended binary primitive BCH codes of length 128 , 1997, IEEE Trans. Inf. Theory.

[96]  Sergio Verdu,et al.  Multiuser Detection , 1998 .

[97]  Paul H. Siegel,et al.  Performance analysis of turbo-equalized partial response channels , 2001, IEEE Trans. Commun..

[98]  Igal Sason,et al.  An Improved Sphere-Packing Bound Targeting Codes of Short to Moderate Block Lengths and Applications , 2006, ArXiv.

[99]  Paul M. Ebert,et al.  Error bounds for parallel communication channels. , 1966 .

[100]  Michael A. Temple,et al.  Approach for deriving performance bounds of punctured turbo codes , 1999 .

[101]  Shlomo Shamai,et al.  Fading channels: How perfect need "Perfect side information" be? , 2002, IEEE Trans. Inf. Theory.

[102]  Dariush Divsalar,et al.  Maximum likelihood decoding analysis of accumulate-repeat-accumulate codes , 2004, IEEE Global Telecommunications Conference, 2004. GLOBECOM '04..

[103]  Emre Telatar,et al.  On the Asymptotic Input-Output Weight Distributions and Thresholds of Convolutional and Turbo-Like Codes , 2006 .

[104]  Kari Pehkonen,et al.  Performance Evaluation of Superorthogonal Turbo Codes in AWGN and Flat Rayleigh Fading Channels , 1998, IEEE J. Sel. Areas Commun..

[105]  Marco Breiling,et al.  A Method for Determining the Distance Pro £ le of Turbo Codes , 2007 .

[106]  Amir K. Khandani,et al.  Generalized tangential sphere bound on the ML decoding error probability of linear binary block codes in AWGN interference , 2004, IEEE Transactions on Information Theory.

[107]  J. R. Cruz,et al.  Bounds for low-density parity-check codes over partial-response channels , 2002 .

[108]  Shlomo Shamai,et al.  Achievable performance over the correlated Rician channel , 1994, IEEE Trans. Commun..

[109]  A. Montanari Turbo codes: the phase transition , 2000, cond-mat/0003218.

[110]  C. Shannon Probability of error for optimal codes in a Gaussian channel , 1959 .

[111]  Yuri V. Svirid Weight distributions and bounds for turbo-codes , 1995, Eur. Trans. Telecommun..

[112]  Krishna R. Narayanan,et al.  Iterative soft decoding of Reed-Solomon codes , 2004, IEEE Communications Letters.

[113]  Sergio Benedetto,et al.  Performance of continuous and blockwise decoded turbo codes , 1997, IEEE Communications Letters.

[114]  Steven W. McLaughlin,et al.  Rate-compatible puncturing of low-density parity-check codes , 2004, IEEE Transactions on Information Theory.

[115]  E.R. Berlekamp,et al.  The technology of error-correcting codes , 1980, Proceedings of the IEEE.

[116]  Igal Sason,et al.  On Achievable Rates and Complexity of LDPC Codes Over Parallel Channels: Bounds and Applications , 2007, IEEE Transactions on Information Theory.

[117]  R. Urbanke,et al.  On the ensemble performance of turbo codes , 1997, Proceedings of IEEE International Symposium on Information Theory.

[118]  S. Baggen,et al.  Union bounds on the performance of product codes , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[119]  Giuseppe Caire,et al.  Bit-Interleaved Coded Modulation , 2008, Found. Trends Commun. Inf. Theory.

[120]  Linda W. Hughes,et al.  A simple upper bound on the error probability for orthogonal signals in white noise , 1992, IEEE Trans. Commun..

[121]  Rudiger Urbanke,et al.  Information-theoretic lower bounds on the bit error probability of codes on graphs , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[122]  T. Duman,et al.  New performance bounds for turbo codes , 1997, GLOBECOM 97. IEEE Global Telecommunications Conference. Conference Record.

[123]  Idan Goldenberg,et al.  Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations , 2006, ArXiv.

[124]  V. B. Balakirsky,et al.  Estimates of the Decoding Error Probability for parallel channels with dependent noise , 2004 .

[125]  Raymond Knopp,et al.  On coding for block fading channels , 2000, IEEE Trans. Inf. Theory.

[126]  Jeremy Thorpe,et al.  Enumerators for protograph ensembles of LDPC codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[127]  Robert J. McEliece,et al.  RA Codes Achieve AWGN Channel Capacity , 1999, AAECC.

[128]  Christian Schlegel,et al.  On error bounds and turbo-codes , 1999, IEEE Communications Letters.

[129]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[130]  R. McEliece,et al.  Bounds on the Average Binary Minimum Distance and the Maximum Likelihood Performance of Reed , 2005 .

[131]  Masoud Salehi,et al.  Performance bounds for turbo-coded modulation systems , 1999, IEEE Trans. Commun..

[132]  Alexander Barg,et al.  On the asymptotic accuracy of the union bound , 2004, ArXiv.

[133]  Uri Erez,et al.  The ML decoding performance of LDPC ensembles over Z/sub q/ , 2005, IEEE Transactions on Information Theory.

[134]  Andrea Montanari,et al.  Tight bounds for LDPC codes under MAP decoding , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[135]  Hyundong Shin,et al.  Improved upper bound on the bit error probability of turbo codes for ML decoding with imperfect CSI in a Rayleigh fading channel , 2001, 12th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications. PIMRC 2001. Proceedings (Cat. No.01TH8598).

[136]  Li Ping,et al.  Performance analysis of turbo-SPC codes , 2004, IEEE Transactions on Information Theory.

[137]  Igal Sason,et al.  On the Parity-Check Density and Achievable Rates of LDPC Codes , 2005, ArXiv.

[138]  Andrea Montanari,et al.  Maxwell Construction: The Hidden Bridge Between Iterative and Maximum a Posteriori Decoding , 2005, IEEE Transactions on Information Theory.

[139]  Gordon L. Stüber,et al.  A serial concatenation approach to iterative demodulation and decoding , 1999, IEEE Trans. Commun..

[140]  Fulvio Babich On the performance of efficient coding techniques over fading channels , 2004, IEEE Transactions on Wireless Communications.

[141]  Amir K. Khandani,et al.  A new upper bound on the ML decoding error probability of linear binary block codes in AWGN interference , 2004, IEEE Transactions on Information Theory.

[142]  M. Namokel Error performance bounds of turbo-codes employing nonuniform interleavers , 1999, 1999 IEEE International Conference on Personal Wireless Communications (Cat. No.99TH8366).

[143]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[144]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[145]  Marc P. C. Fossorier,et al.  Sphere-packing bounds revisited for moderate block lengths , 2004, IEEE Transactions on Information Theory.

[146]  Brian L. Hughes,et al.  On the error probability of signals in additive white Gaussian noise , 1991, IEEE Trans. Inf. Theory.

[147]  Idan Goldenberg,et al.  Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes , 2007, IEEE Transactions on Information Theory.

[148]  Dariush Divsalar,et al.  Transfer function bounds on the performance of turbo codes Pasadena , 1995 .

[149]  Shu Lin,et al.  On the weight distribution of terminated convolutional codes , 1999, IEEE Trans. Inf. Theory.

[150]  G. Battail,et al.  The Random Coded Modulation: Performance And Euclidean Distance Spectrum Evaluation , 1991, Proceedings. 1991 IEEE International Symposium on Information Theory.

[151]  J. Galambos,et al.  Bonferroni-type inequalities with applications , 1996 .

[152]  L. Bazzi,et al.  Exact thresholds and optimal codes for the binary symmetric channel and Gallager's decoding algorithm A , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[153]  N. Sloane,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. I , 1993 .

[154]  Andrew J. Viterbi,et al.  Principles of coherent communication , 1966 .

[155]  Neri Merhav,et al.  Lower bounds on the error probability of block codes based on improvements on de Caen's inequality , 2004, IEEE Transactions on Information Theory.

[156]  Shlomo Shamai,et al.  Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[157]  Weihua Zhuang,et al.  Variance of the turbo-code performance bound over the interleavers , 1999, 1999 IEEE 49th Vehicular Technology Conference (Cat. No.99CH36363).

[158]  Shlomo Shamai,et al.  Fading channels (invited paper): information-theoretic and communications aspects , 2000 .

[159]  Daniel J. Costello,et al.  An algorithm for computing the distance spectrum of trellis codes , 1989, IEEE Journal on Selected Areas in Communications.

[160]  Robert F. H. Fischer,et al.  Multilevel codes: Theoretical concepts and practical design rules , 1999, IEEE Trans. Inf. Theory.

[161]  Dariush Divsalar,et al.  Serial and Hybrid Concatenated Codes with Applications , 1997 .

[162]  Peter F. Swaszek A lower bound on the error probability for signals in white Gaussian noise , 1995, IEEE Trans. Inf. Theory.

[163]  Shlomo Shamai,et al.  Variations on the Gallager bounds, connections, and applications , 2002, IEEE Trans. Inf. Theory.

[164]  Toru Fujiwara,et al.  Determination of the Local Weight Distribution of Binary Linear Block Codes , 2006, IEEE Transactions on Information Theory.

[165]  Robert G. Gallager,et al.  A simple derivation of the coding theorem and some applications , 1965, IEEE Trans. Inf. Theory.

[166]  Ghassan Kawas Kaleh,et al.  Chernoff bound of trellis-coded modulation over correlated fading channels , 1994, IEEE Trans. Commun..

[167]  Andrea Montanari,et al.  Weight distributions of LDPC code ensembles: combinatorics meets statistical physics , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[168]  Emina Soljanin,et al.  Reliable channel regions for good binary codes transmitted over parallel channels , 2006, IEEE Transactions on Information Theory.

[169]  David Burshtein,et al.  Asymptotic enumeration methods for analyzing LDPC codes , 2004, IEEE Transactions on Information Theory.

[170]  Andrea Montanari,et al.  Life Above Threshold: From List Decoding to Area Theorem and MSE , 2004, ArXiv.

[171]  Chintha Tellambura,et al.  Generation of bivariate Rayleigh and Nakagami-m fading envelopes , 2000, IEEE Communications Letters.

[172]  Øyvind Ytrehus,et al.  Improved algorithms for the determination of turbo-code weight distributions , 2005, IEEE Transactions on Communications.

[173]  Toru Fujiwara,et al.  Relations between the local weight distributions of a linear block code, its extended code, and its even weight subcode , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[174]  E. Kurtas,et al.  Performance bounds for high rate linear codes over partial response channels , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[175]  Tolga M. Duman,et al.  Maximum likelihood decoding bounds for high rate turbo codes over Lorentzian channels , 2001, ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240).

[176]  A. G. Burr,et al.  Bounds on coding gain versus decoding delay for spherical codes on the Gaussian channel , 1997, Proceedings of IEEE International Symposium on Information Theory.

[177]  Tolga M. Duman,et al.  Maximum likelihood decoding bounds for high rate turbo codes over Lorentzian channels , 2001 .

[178]  Dariush Divsalar,et al.  Analysis, Design, and Iterative Decoding of Double Serially Concatenated Codes with Interleavers , 1998, IEEE J. Sel. Areas Commun..

[179]  Erik Agrell,et al.  On the Voronoi Neighbor Ratio for Binary Linear Block Codes , 1998, IEEE Trans. Inf. Theory.

[180]  Fady Alajaji,et al.  Improved Lower Bounds for the Error Rate of Linear Block Codes , 2005 .

[181]  Paul H. Siegel,et al.  Serial concatenated TCM with an inner accumulate code-part I: maximum-likelihood analysis , 2005, IEEE Transactions on Communications.

[182]  D. Burshtein,et al.  Upper bounds on the rate of LDPC codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[183]  Ludo M. G. M. Tolhuizen,et al.  On the weight enumerator of product codes , 1992, Discret. Math..

[184]  Tomoaki Ohtsuki,et al.  Transfer function bounds on performance of binary turbo-coding followed by M-ary orthogonal signal mapping through interleaver , 2000, 2000 IEEE International Conference on Communications. ICC 2000. Global Convergence Through Communications. Conference Record.

[185]  Dariush Divsalar,et al.  Upper bounds to error probabilities of coded systems beyond the cutoff rate , 2003, IEEE Trans. Commun..

[186]  G. David Forney,et al.  Modulation and Coding for Linear Gaussian Channels , 1998, IEEE Trans. Inf. Theory.

[187]  Scott L. Miller,et al.  An upper bound on turbo codes performance over quasi-static fading channels , 2003, IEEE Communications Letters.

[188]  Sergio Benedetto,et al.  Unveiling turbo codes: some results on parallel concatenated coding schemes , 1996, IEEE Trans. Inf. Theory.

[189]  S. G. Wilson,et al.  Design and analysis of turbo codes on Rayleigh fading channels , 1996, Proceedings of GLOBECOM'96. 1996 IEEE Global Telecommunications Conference.

[190]  Shlomo Shamai,et al.  Improved upper bounds on the ML decoding error probability of parallel and serial concatenated turbo codes via their ensemble distance spectrum , 2000, IEEE Trans. Inf. Theory.

[191]  Ludo M. G. M. Tolhuizen More results on the weight enumerator of product codes , 2002, IEEE Trans. Inf. Theory.

[192]  David Burshtein,et al.  On the application of LDPC codes to arbitrary discrete-memoryless channels , 2003, IEEE Transactions on Information Theory.

[193]  Shlomo Shamai,et al.  Improved upper bounds on the ensemble performance of ML decoded low density parity check codes , 2000, IEEE Communications Letters.

[194]  Cong Ling,et al.  New Gallager Bounds in Block-Fading Channels , 2007, IEEE Transactions on Information Theory.

[195]  Dariush Divsalar,et al.  Ensemble Weight Enumerators for Protograph LDPC Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[196]  Gregory Poltyrev,et al.  The error probability of M-ary PSK block coded modulation schemes , 1996, IEEE Trans. Commun..

[197]  Alexander Barg,et al.  Minimal Vectors in Linear Codes , 1998, IEEE Trans. Inf. Theory.

[198]  O. Collins,et al.  A comparison of known codes, random codes, and the best codes , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[199]  Andrea Montanari,et al.  Tight bounds for LDPC and LDGM codes under MAP decoding , 2004, IEEE Transactions on Information Theory.

[200]  Erik Agrell,et al.  Voronoi regions for binary linear block codes , 1996, IEEE Trans. Inf. Theory.

[201]  S. Dolinar,et al.  Weight distributions for turbo codes using random and nonrandom permutations , 1995 .

[202]  Kamil Sh. Zigangirov,et al.  Tighter bounds on the error probability of fixed convolutional codes , 2001, IEEE Trans. Inf. Theory.

[203]  Steven W. McLaughlin,et al.  On performance limits of space-time codes: a sphere-packing bound approach , 2003, IEEE Trans. Inf. Theory.

[204]  Dariush Divsalar,et al.  Code Performance as a Function of Block Size , 1998 .

[205]  S. Shamai,et al.  Information rates and error exponents of compound channels with application to antipodal signaling in a fading environment , 1993 .

[206]  Giuseppe Caire,et al.  Coded modulation in the block-fading channel: coding theorems and code construction , 2006, IEEE Transactions on Information Theory.

[207]  Dominique de Caen,et al.  A lower bound on the probability of a union , 1997, Discret. Math..

[208]  Paul H. Siegel,et al.  The serial concatenation of rate-1 codes through uniform random interleavers , 2003, IEEE Trans. Inf. Theory.

[209]  Jing Li,et al.  Generalized product accumulate codes: analysis and performance , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[210]  Shlomo Shamai,et al.  Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels , 2003, IEEE Trans. Commun..

[211]  Johannes Zangl,et al.  Improved tangential sphere bound on the bit-error probability of concatenated codes , 2001, IEEE J. Sel. Areas Commun..

[212]  Joseph Jean Boutros,et al.  Serial concatenation of interleaved convolutional codes and M -ary continuous phase modulations - Concaténation série de codes convolutifs et de modulations à phase continue m -aires séparés par un entrelaceur , 1999, Ann. des Télécommunications.

[213]  En-Hui Yang,et al.  On the input-output weight enumerators of product accumulate codes , 2004, IEEE Communications Letters.