Journal Home > Volume 28 , Issue 2

In this paper, we propose the two-stage constructions for the rate-compatible shortened polar (RCSP) codes. For the Stage-I construction, the shortening pattern and the frozen bit are jointly designed to make the shortened bits be completely known by the decoder. Besides, a distance-greedy algorithm is presented to improve the minimum Hamming distance of the codes. To design the remaining Stage-II frozen bits, three different construction algorithms are further presented, called the Reed-Muller (RM) construction, the Gaussian Approximation (GA) construction, and the RM-GA construction. Then we give the row weight distribution numerical results of the generator matrix after the Stage-I and Stage-II constructions, which shows that the proposed constructions can efficiently increase the minimum Hamming distance. Simulation results show that the proposed RCSP codes have excellent frame error rate (FER) performances at different code lengths and code rates. More specifically, the RM-GA construction performs best and can achieve at most 0.8 dB gain compared to the Wang14 and the quasi-uniform puncturing (QUP) schemes. The RM construction is designed completely by the distance-constraint without channel evaluation thus has the simplest structure. Interestingly, it still has better FER performance than the existing shortening/puncturing schemes, especially at high signal noise ratio (SNR) region.


menu
Abstract
Full text
Outline
About this article

Two-Stage Constructions for the Rate-Compatible Shortened Polar Codes

Show Author's information Chunjie Li1Haiqiang Chen1,2( )Zelin Wang1Youming Sun1Xiangcheng Li1Tuanfa Qin1
School of Computer, Electronics and Information and Guangxi Colleges and Universities Key Laboratory of Multimedia Communications and Information Processing, Guangxi University, Nanning 530004, China
Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi University, Nanning 530004, China

Abstract

In this paper, we propose the two-stage constructions for the rate-compatible shortened polar (RCSP) codes. For the Stage-I construction, the shortening pattern and the frozen bit are jointly designed to make the shortened bits be completely known by the decoder. Besides, a distance-greedy algorithm is presented to improve the minimum Hamming distance of the codes. To design the remaining Stage-II frozen bits, three different construction algorithms are further presented, called the Reed-Muller (RM) construction, the Gaussian Approximation (GA) construction, and the RM-GA construction. Then we give the row weight distribution numerical results of the generator matrix after the Stage-I and Stage-II constructions, which shows that the proposed constructions can efficiently increase the minimum Hamming distance. Simulation results show that the proposed RCSP codes have excellent frame error rate (FER) performances at different code lengths and code rates. More specifically, the RM-GA construction performs best and can achieve at most 0.8 dB gain compared to the Wang14 and the quasi-uniform puncturing (QUP) schemes. The RM construction is designed completely by the distance-constraint without channel evaluation thus has the simplest structure. Interestingly, it still has better FER performance than the existing shortening/puncturing schemes, especially at high signal noise ratio (SNR) region.

Keywords: polar codes, rate-compatibility, Reed-Muller codes, Hamming distance, shortening

References(30)

[1]
E. Arikan, Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels, IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, 2009.
[2]
K. Chen, K. Niu, and J. R. Lin, List successive cancellation decoding of polar codes, Electron. Lett., vol. 48, no. 9, pp. 500–501, 2012.
[3]
TS38.212 V.15.1.0, NR; Multiplexing and channel coding, https://www.3gpp.org/DynaReport/38-series.htm, 2018.
[4]
C. Sun, Z. S. Fei, D. Jia, C. Z. Cao, and X. Y. Wang, Secure transmission scheme for parallel relay channels based on polar coding, Tsinghua Science and Technology, vol. 23, no. 3, pp. 357–365, 2018.
[5]
J. Du, J. Song, Y. Ren, and J. T. Wang, Convergence of broadband and broadcast/multicast in maritime information networks, Tsinghua Science and Technology, vol. 26, no. 5, pp. 592–607, 2021.
[6]
TS38.212, V15.0.0, NR; Multiplexing and channel coding, https://www.3gpp.org/DynaReport/38-series.htm, 2017.
[7]
F. Gabry, V. Bioglio, I. Land, and J. C. Belfiore, Multi-kernel construction of polar codes, in Proc. 2017 IEEE Int. Conf. on Communications Workshops (ICC Workshops), Paris, France, 2017, pp. 761–765.
[8]
A. Eslami and H. Pishro-Nik, A practical approach to polar codes, in Proc. 2011 IEEE Int. Symp. on Information Theory Proc., St. Petersburg, Russia, 2011, pp. 16–20.
[9]
K. Niu, K. Chen, and J. R. Lin, Beyond turbo codes: Rate-compatible punctured polar codes, in Proc. 2013 IEEE Int. Conf. on Communications (ICC), Budapest, Hungary, 2013, pp. 3423–3427.
[10]
R. X. Wang and R. K. Liu, A novel puncturing scheme forpolar codes, IEEE Commun. Lett., vol. 18, no. 12, pp. 2081–2084, 2014.
[11]
K. Niu, J. C. Dai, K. Chen, J. R. Lin, Q. T. Zhang, and A. V. Vasilakos, Rate-compatible punctured polar codes: Optimal construction based on polar spectra, arXiv preprint arXiv: 1612.01352, 2016.
[12]
B. Li, H. Shen, and D. Tse, An adaptive successive cancellation list decoder for polar codes with cyclic redundancy check, IEEE Commun. Lett., vol. 16, no. 12, pp. 2044–2047, 2012.
[13]
V. Miloslavskaya, Shortened polar codes, IEEE Trans. Inf. Theory, vol. 61, no. 9, pp. 4852–4865, 2015.
[14]
B. Li, H. Shen, and D. Tse, A RM-polar codes, arXiv preprint arXiv: 1407.5483, 2014.
[15]
M. Bakshi, S. Jaggi, and M. Effros, Concatenated polar codes, in Proc. 2010 IEEE Int. Symp. on Information Theory, Austin, TX, USA, 2010, pp. 918–922.
[16]
I. Tal and A. Vardy, List decoding of polar codes, in Proc. 2011 IEEE Int. Symp. on Information Theory Proc., St. Petersburg, Russia, 2011, pp. 1–5.
[17]
K. Niu and K. Chen, CRC-aided decoding of polar codes, IEEE Commun. Lett., vol. 16, no. 10, pp. 1668–1671, 2012.
[18]
E. Arikan, Serially concatenated polar codes, IEEE Access, vol. 6, pp. 64549–64555, 2018.
[19]
T. Wang, D. M. Qu, and T. Jiang, Parity-check-concatenated polar codes, IEEE Commun. Lett., vol. 20, no. 12, pp. 2342–2345, 2016.
[20]
R. Gallage, Low-density parity-check codes, IRE Trans. Inf. Theory, vol. 8, no. 1, pp. 21–28, 1962.
[21]
P. Wang, L. G. Yin, and J. H. Lu, Efficient helicopter-satellite communication scheme based on check-hybrid LDPC coding, Tsinghua Science and Technology, vol. 23, no. 3, pp. 323–332, 2018.
[22]
B. H. Lin, Y. K. Pei, L. G. Yin, and J. H. Lu, Design and efficient hardware implementation schemes for non-Quasi-Cyclic LDPC codes, Tsinghua Science and Technology, vol. 22, no. 1, pp. 92–103, 2017.
[23]
C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1, in Proc. ICC’93–IEEE Int. Conf. on Communications, Geneva, Switzerland, 1993, pp. 1064–1070.
[24]
D. E. Muller, Application of Boolean algebra to switching circuit design and to error detection, Trans.I.R.E. Prof. Group Electron. Comput., vol. EC-3, no. 3, pp. 6–12, 1954.
[25]
I. Reed, A class of multiple-error-correcting codes and the decoding scheme, Trans. IRE Prof. Group on Inf. Theory, vol. 4, no. 4, pp. 38–49, 1954.
[26]
E. Arikan, A survey of reed-muller codes from polar coding perspective, in Proc. 2010 IEEE Information Theory Workshop on Information Theory (ITW), Cairo, Egypt, 2010, pp. 1–5.
[27]
R. Mori and T. Tanaka, Performance of polar codes with the construction using density evolution, IEEE Commun. Lett., vol. 13, no. 7, pp. 519–521, 2009.
[28]
P. Trifonov, Efficient design and decoding of polar codes, IEEE Trans. Commun., vol. 60, no. 11, pp. 3221–3227, 2012.
[29]
G. N. He, J. C. Belfiore, I. Land, G. H. Yang, X. C. Liu, Y. Chen, R. Li, J. Wang, Y. Q. Ge, R. Zhang, et al., Beta-expansion: A theoretical framework for fast and recursive construction of polar codes, in Proc. 2017 IEEE Global Communications Conf., Singapore, 2017, pp. 1–6.
[30]
V. Bioglio, F. Gabry, and I. Land, Low-complexity puncturing and shortening of polar codes, in Proc. 2017 IEEE Wireless Communications and Networking Conf. Workshops (WCNCW), San Francisco, CA, USA, 2017, pp. 1–6.
Publication history
Copyright
Acknowledgements
Rights and permissions

Publication history

Received: 01 September 2021
Revised: 04 November 2021
Accepted: 05 November 2021
Published: 29 September 2022
Issue date: April 2023

Copyright

© The author(s) 2023.

Acknowledgements

The authors are very grateful to the editor and reviewers for their comments and constructive suggestions, which help to enrich the content and improve the presentation of this paper. This work was supported by the Interdisciplinary Scientific Research Foundation of GuangXi University (No. 2022JCC015), the National Natural Science Foundation of China (Nos. 61761006, 61961004, and 61762011), and the Natural Science Foundation of Guangxi of China (Nos. 2017GXNSFAA198263 and 2018GXNSFAA294059).

Rights and permissions

The articles published in this open access journal are distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).

Return