Polar Subcodes

An extension of polar codes is proposed, which allows some of the frozen symbols, called dynamic frozen symbols, to be data-dependent. A construction of polar codes with dynamic frozen symbols, being subcodes of extended BCH codes, is proposed. The proposed codes have higher minimum distance than classical polar codes, but still can be efficiently decoded using the successive cancellation algorithm and its extensions. The codes with Arikan, extended BCH and Reed-Solomon kernel are considered. The proposed codes are shown to outperform LDPC and turbo codes, as well as polar codes with CRC.

[1]  Santhosh Kumar,et al.  Reed-Muller Codes Achieve Capacity on Erasure Channels , 2017, IEEE Trans. Inf. Theory.

[2]  Ying Wang,et al.  Concatenations of polar codes with outer BCH codes and convolutional codes , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[3]  Raouf Hamzaoui,et al.  Fast tree-trellis list Viterbi decoding , 2006, IEEE Transactions on Communications.

[4]  Alexander Vardy,et al.  Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes , 2016, IEEE Transactions on Communications.

[5]  Vera Miloslavskaya,et al.  Sequential decoding of Reed-Solomon codes , 2014, 2014 International Symposium on Information Theory and its Applications.

[6]  Rüdiger L. Urbanke,et al.  Polar Codes for Channel and Source Coding , 2009, ArXiv.

[7]  Peter Trifonov,et al.  Binary successive cancellation decoding of polar codes with Reed-Solomon kernel , 2014, 2014 IEEE International Symposium on Information Theory.

[8]  Jean-Marie Goethals,et al.  On Generalized Reed-Muller Codes and Their Relatives , 1970, Inf. Control..

[9]  Rüdiger L. Urbanke,et al.  On the scaling of polar codes: I. The behavior of polarized channels , 2010, 2010 IEEE International Symposium on Information Theory.

[10]  Tadao Kasami,et al.  New generalizations of the Reed-Muller codes-I: Primitive codes , 1968, IEEE Trans. Inf. Theory.

[11]  Mayank Bakshi,et al.  Concatenated Polar codes , 2010, 2010 IEEE International Symposium on Information Theory.

[12]  Shu Lin,et al.  On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes , 1993, IEEE Trans. Inf. Theory.

[13]  Peter Trifonov.  Successive cancellation permutation decoding of Reed-Solomon codes , 2014, 2014 IEEE Information Theory Workshop (ITW 2014).

[14]  Rüdiger L. Urbanke,et al.  Polar Codes: Characterization of Exponent, Bounds, and Constructions , 2010, IEEE Transactions on Information Theory.

[15]  K. Niu,et al.  Stack decoding of polar codes , 2012 .

[16]  B. Liesenfeld,et al.  On the Equivalence of Some Generalized Concatenated Codes and Extended Cyclic Codes , 1993, Proceedings. IEEE International Symposium on Information Theory.

[17]  Paul H. Siegel,et al.  Enhanced belief propagation decoding of polar codes through concatenation , 2014, 2014 IEEE International Symposium on Information Theory.

[18]  Rüdiger L. Urbanke,et al.  From polar to Reed-Muller codes: A technique to improve the finite-length performance , 2014, 2014 IEEE International Symposium on Information Theory.

[19]  Vera Miloslavskaya,et al.  Sequential decoding of polar codes with arbitrary binary kernel , 2014, 2014 IEEE Information Theory Workshop (ITW 2014).

[20]  Marc P. C. Fossorier,et al.  Box and match techniques applied to soft-decision decoding , 2002, IEEE Transactions on Information Theory.

[21]  Peter Trifonov,et al.  Efficient Design and Decoding of Polar Codes , 2012, IEEE Transactions on Communications.

[22]  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.

[23]  Alexander Vardy,et al.  Maximum-Likelihood Soft Decision Decoding of Bch Codes , 1993, Proceedings. IEEE International Symposium on Information Theory.

[24]  Kai Chen,et al.  Improved Successive Cancellation Decoding of Polar Codes , 2012, IEEE Transactions on Communications.

[25]  Vera Miloslavskaya,et al.  Polar codes with dynamic frozen symbols and their decoding by directed search , 2013, 2013 IEEE Information Theory Workshop (ITW).

[26]  Peter Trifonov.  Successive cancellation decoding of Reed-Solomon codes , 2014, Probl. Inf. Transm..

[27]  Bin Li,et al.  An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check , 2012, IEEE Communications Letters.

[28]  Peter Trifonov,et al.  Block sequential decoding of polar codes , 2015, 2015 International Symposium on Wireless Communication Systems (ISWCS).

[29]  Hideki Imai,et al.  A new multilevel coding method using error-correcting codes , 1977, IEEE Trans. Inf. Theory.

[30]  Toshiyuki Tanaka,et al.  Rate-Dependent Analysis of the Asymptotic Behavior of Channel Polarization , 2011, IEEE Transactions on Information Theory.

[31]  Alexander Vardy,et al.  How to Construct Polar Codes , 2011, IEEE Transactions on Information Theory.

[32]  Kai Chen,et al.  CRC-Aided Decoding of Polar Codes , 2012, IEEE Communications Letters.

[33]  Toshiyuki Tanaka,et al.  Source and Channel Polarization Over Finite Fields and Reed–Solomon Matrices , 2012, IEEE Transactions on Information Theory.

[34]  Rüdiger L. Urbanke,et al.  Reed-Muller Codes Achieve Capacity on the Binary Erasure Channel under MAP Decoding , 2015, ArXiv.

[35]  Shu Lin,et al.  Suboptimum decoding of decomposable block codes , 1994, IEEE Trans. Inf. Theory.

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

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

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

[39]  Vera Miloslavskaya,et al.  Sequential Decoding of Polar Codes , 2014, IEEE Communications Letters.

[40]  Jungwon Lee,et al.  On the construction and decoding of concatenated polar codes , 2013, 2013 IEEE International Symposium on Information Theory.

[41]  Santhosh Kumar,et al.  Reed–Muller Codes Achieve Capacity on Erasure Channels , 2015, IEEE Transactions on Information Theory.

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

[43]  Vera Miloslavskaya,et al.  Design of binary polar codes with arbitrary kernel , 2012, 2012 IEEE Information Theory Workshop.

[44]  Ilya Dumer,et al.  Soft-decision decoding of Reed-Muller codes: recursive lists , 2006, IEEE Transactions on Information Theory.

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