Sphere decoder for polar codes concatenated with cyclic redundancy check

Abstract

The existing cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL) decoder partitions the decoding process into two steps, where an SCL is followed by a CRC check. An SCL decoder can approach the maximum-likelihood (ML) decoding performance of the inner polar codes using a sufficiently large list; however, in this case, CRC is only used for performing error detection. Therefore, the decoding performance of the outer CRC is different from that of ML because the errors are not rectified, which degrades the performance of the entire concatenated codes. In this study, we propose a sphere decoder (SD) that can achieve the ML performance of polar codes concatenated with CRC to address the suboptimality of CA-SCL decoding. The proposed SD performs joint decoding of the CRC-polar codes in a single step, thereby avoiding the polar decoding and CRC detection decoding scheme. Because the proposed SD guarantees the ML decoding performance of the CRC-polar concatenated codes, the simulation results demonstrate that the block error rate (BLER) of the proposed SD acts as the lower bound of the CA-SCL decoding performance. Further, a new initial radius selection method is proposed for the SD to reduce the average decoding complexity. The simulations demonstrate that the proposed initial radius selection method reduces more amount of decoding complexity when compared with that reduced using sequential step size methods.

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References

  1. 1

    Arikan E. Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Trans Inform Theor, 2009, 55: 3051–3073

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Tal I, Vardy A. List decoding of polar codes. IEEE Trans Inform Theor, 2015, 61: 2213–2226

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Niu K, Chen K. CRC-aided decoding of polar codes. IEEE Commun Lett, 2012, 16: 1668–1671

    Article  Google Scholar 

  4. 4

    Tahir B, Schwarz S, Rupp M. BER comparison between convolutional, turbo, LDPC, and polar codes. In: Proceedings of the 24th International Conference on Telecommunications (ICT), Limassol, 2017. 1–7

    Google Scholar 

  5. 5

    Cao C, Kuang J, Fei Z, et al. Low complexity list successive cancellation decoding of polar codes. IET Commun, 2014, 8: 3145–3149

    Article  Google Scholar 

  6. 6

    Xu Q Y, Pan Z W, Liu N, et al. A complexity-reduced fast successive cancellation list decoder for polar codes. Sci China Inf Sci, 2018, 61: 022309

    MathSciNet  Article  Google Scholar 

  7. 7

    Ryan W E, Lin S. Channel Codes Classical And Modern. New York: Cambridge University Press Ltd., 2009. 110–111

    Google Scholar 

  8. 8

    Kazakov P. Fast calculation of the number of minimum-weight words of CRC codes. IEEE Trans Inform Theor, 2001, 47: 1190–1195

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Castagnoli G, Brauer S, Herrmann M. Optimization of cyclic redundancy-check codes with 24 and 32 parity bits. IEEE Trans Commun, 1993, 41: 883–892

    Article  MATH  Google Scholar 

  10. 10

    Kahraman S, Celebi M E. Code based efficient maximum-likelihood decoding of short polar codes. In: Proceedings of 2012 IEEE International Symposium on Information Theory (ISIT), Cambridge, 2012. 1967–1971

    Google Scholar 

  11. 11

    Niu K, Chen K, Lin J. Low-complexity sphere decoding of polar codes based on optimum path metric. IEEE Commun Lett, 2014, 18: 332–335

    Article  Google Scholar 

  12. 12

    Guo J, Fabregas A G. Efficient sphere decoding of polar codes. In: Proceedings of IEEE International Symposium on Information Theory (ISIT), Hong Kong, 2015. 236–240

    Google Scholar 

  13. 13

    Hashemi S A, Condo C, Gross W J. List sphere decoding of polar codes. In: Proceedings of the 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, 2015. 1346–1350

    Google Scholar 

  14. 14

    Hashemi S A, Condo C, Gross W J. Matrix reordering for efficient list sphere decoding of polar codes. In: Proceedings of 2016 IEEE International Symposium on Circuits and Systems (ISCAS), Montreal, 2016. 1730–1733

    Google Scholar 

  15. 15

    Husmann C, Nikolaou P C, Nikitopoulos K. Reduced latency ML polar decoding via multiple sphere-decoding tree searches. IEEE Trans Veh Technol, 2018, 67: 1835–1839

    Article  Google Scholar 

  16. 16

    Trifonov P. Efficient design and decoding of polar codes. IEEE Trans Commun, 2012, 60: 3221–3227

    Article  Google Scholar 

  17. 17

    Tal I, Vardy A. How to construct polar codes. IEEE Trans Inform Theor, 2013, 59: 6562–6582

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    He G, Belfiore J C, Land I, et al. Beta-expansion: a theoretical framework for fast and recursive construction of polar codes. In: Proceedings of IEEE Global Communications Conference, Singapore, 2017. 1–6

    Google Scholar 

  19. 19

    Koopman P, Chakravarty T. Cyclic redundancy code (CRC) polynomial selection for embedded networks. In: Proceedings of International Conference on Dependable Systems and Networks, Florence, 2004. 145–154

    Google Scholar 

  20. 20

    Agrell E, Eriksson T, Vardy A, et al. Closest point search in lattices. IEEE Trans Inform Theor, 2002, 48: 2201–2214

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Zhang Q, Liu A, Pan X, et al. CRC code design for list decoding of polar codes. IEEE Commun Lett, 2017, 21: 1229–1232

    Article  Google Scholar 

  22. 22

    Hassibi B, Vikalo H. On the sphere-decoding algorithm I. Expected complexity. IEEE Trans Signal Process, 2005, 53: 2806–2818

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Zhao W, Giannakis G B. Sphere decoding algorithms with improved radius search. IEEE Trans Commun, 2005, 53: 1104–1109

    Article  Google Scholar 

  24. 24

    Balatsoukas-Stimming A, Parizi M B, Burg A. LLR-based successive cancellation list decoding of polar codes. IEEE Trans Signal Process, 2015, 63: 5165–5179

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Zhiwen Pan.

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Yu, Y., Pan, Z., Liu, N. et al. Sphere decoder for polar codes concatenated with cyclic redundancy check. Sci. China Inf. Sci. 62, 82303 (2019). https://doi.org/10.1007/s11432-018-9743-0

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Keywords

  • polar codes
  • sphere decoder
  • maximum-likelihood decoding
  • optimal decoding
  • radius search