Successive cancellation decoding of single parity-check product codes

We introduce successive cancellation (SC) decoding of product codes (PCs) with single parity-check (SPC) component codes. Recursive formulas are derived, which resemble the SC decoding algorithm of polar codes. We analyze the error probability of SPC-PCs over the binary erasure channel under SC decoding. A bridge with the analysis of PCs introduced by Elias in 1954 is also established. Furthermore, bounds on the block error probability under SC decoding are provided, and compared to the bounds under the original decoding algorithm proposed by Elias. It is shown that SC decoding of SPC-PCs achieves a lower block error probability than Elias' decoding.

[1]  Giuseppe Durisi,et al.  List Decoding of Short Codes for Communication over Unknown Fading Channels , 2019, 2019 53rd Asilomar Conference on Signals, Systems, and Computers.

[2]  Michael Lentmaier,et al.  Braided Block Codes , 2009, IEEE Transactions on Information Theory.

[3]  Ryuhei Mori,et al.  Performance and construction of polar codes on symmetric binary-input memoryless channels , 2009, 2009 IEEE International Symposium on Information Theory.

[4]  Alexander Vardy,et al.  List decoding of polar codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[5]  R. Pyndiah,et al.  An overview of turbo codes and their applications , 2005, The European Conference on Wireless Technology, 2005..

[6]  Henry D. Pfister,et al.  Density Evolution for Deterministic Generalized Product Codes on the Binary Erasure Channel at High Rates , 2015, IEEE Transactions on Information Theory.

[7]  Valerio Bioglio,et al.  Multi-kernel polar codes: Proof of polarization and error exponents , 2017, 2017 IEEE Information Theory Workshop (ITW).

[8]  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,.

[9]  Eren Sasoglu,et al.  Polarization and Polar Codes , 2012, Found. Trends Commun. Inf. Theory.

[10]  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.

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

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

[13]  Rüdiger L. Urbanke,et al.  Polar codes: Characterization of exponent, bounds, and constructions , 2009, 2009 IEEE International Symposium on Information Theory.

[14]  Joachim Neu,et al.  Ternary Quantized Polar Code Decoders: Analysis and Design , 2019, 2019 53rd Asilomar Conference on Signals, Systems, and Computers.

[15]  F. Chiaraluce,et al.  Extended Hamming product codes analytical performance evaluation for low error rate applications , 2004, IEEE Transactions on Wireless Communications.

[16]  Gerhard Fettweis,et al.  From product codes to structured generalized LDPC codes , 2010, 2010 5th International ICST Conference on Communications and Networking in China.

[17]  R. Koetter,et al.  Performance of Iterative Algebraic Decoding of Codes Defined on Graphs: An Initial Investigation , 2007, 2007 IEEE Information Theory Workshop.

[18]  Ramesh Pyndiah,et al.  Near-optimum decoding of product codes: block turbo codes , 1998, IEEE Trans. Commun..

[19]  Henry D. Pfister,et al.  Symmetric product codes , 2015, 2015 Information Theory and Applications Workshop (ITA).

[20]  Jean-Claude Belfiore,et al.  Multi-kernel construction of polar codes , 2016, 2017 IEEE International Conference on Communications Workshops (ICC Workshops).

[21]  A. Robert Calderbank,et al.  Beyond double transitivity: Capacity-achieving cyclic codes on erasure channels , 2016, 2016 IEEE Information Theory Workshop (ITW).

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

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

[24]  P. A. Wintz,et al.  Error Free Coding , 1973 .

[25]  Gianluigi Liva,et al.  Successive Cancellation List Decoding of Single Parity-Check Product Codes , 2017 .

[26]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[27]  H. Vincent Poor,et al.  Channel Coding Rate in the Finite Blocklength Regime , 2010, IEEE Transactions on Information Theory.

[28]  Shlomo Shamai,et al.  Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial , 2006, Found. Trends Commun. Inf. Theory.

[29]  M. El-Khamy,et al.  The average weight enumerator and the maximum likelihood performance of product codes , 2005, 2005 International Conference on Wireless Networks, Communications and Mobile Computing.

[30]  Norbert Stolte,et al.  Rekursive Codes mit der Plotkin-Konstruktion und ihre Decodierung , 2002 .

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

[32]  Emre Telatar,et al.  Finite-length analysis of low-density parity-check codes on the binary erasure channel , 2002, IEEE Trans. Inf. Theory.

[33]  Marco Chiani,et al.  Bounds on the Error Probability of Block Codes over the q-Ary Erasure Channel , 2013, IEEE Transactions on Communications.

[34]  Desmond P. Taylor,et al.  Asymptotic performance of single parity-check product codes , 2003, IEEE Trans. Inf. Theory.

[35]  N. J. A. Sloane,et al.  Generalizations of Gleason's theorem on weight enumerators of self-dual codes , 1972, IEEE Trans. Inf. Theory.

[36]  William E. Ryan,et al.  Efficient Error-Correcting Codes in the Short Blocklength Regime , 2018, Phys. Commun..

[37]  Gianluigi Liva,et al.  Successive Cancellation List Decoding of Product Codes With Reed-Muller Component Codes , 2019, IEEE Communications Letters.

[38]  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.

[39]  Henry D. Pfister,et al.  Approaching Miscorrection-Free Performance of Product Codes With Anchor Decoding , 2018, IEEE Transactions on Communications.

[40]  John Cocke,et al.  Optimal decoding of linear codes for minimizing symbol error rate (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[41]  Norman Abramson,et al.  Cascade Decoding of Cyclic Product Codes , 1968 .

[42]  Erdal Arikan,et al.  Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels , 2008, IEEE Transactions on Information Theory.

[43]  T. Aaron Gulliver,et al.  Single parity check product codes , 2001, IEEE Trans. Commun..

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

[45]  Peter Trifonov,et al.  Generalized concatenated codes based on polar codes , 2011, 2011 8th International Symposium on Wireless Communication Systems.