Polar codes: from theory to practice
Project leader: prof. P. Trifonov
Polar codes
Channel polarization phenomenon was discovered by E. Arikan in 2008. Essentially, by employing very simple operations one can transform a data transmission channel into a number almost noiseless and almost pure-noise synthetic subchannels. Then one can transmit the payload data over almost noiseless subchannels with very high reliability. Some deterministic symbols (normally, 0) should be trasmitted over almost pure-noise synthetic suchannels.
This approach defines a forward error correction method (polar coding). E. Arikan has shown that polar codes achieve Shannon capacity of a wide class of communication channels, having very simple construction, encoding and decoding algorithms. This is the first result of such kind since 1948, when C. Shannon shown the theoretical possibility of creation of such coding methods. Although code constructions achieving the capacity of some channels have been previously known, they did not find practical applications due to high implementation complexity. Owing to their simplicity, polar codes have great potential for finding their way into communication and storage systems.
However, it appears that the performance of Arikan polar codes with practically important parameters is substantially inferior compared to similar LDPC and turbo codes. Furthermore, the complexity and latency of classical decoding methods for polar codes significantly exceeds that of LDPC codes. Since 2011, Coding Theory Lab of SPbPU is working on overcoming these problems and development of code constructions and decoding algorithms with better performance and lower complexity compared to the existing techniques.
Polar subcodes (Polar codes with dynamic frozen symbols)
P. V. Trifonov and V. D. Miloslavskaya have suggested a generalization of the polar code construction. Instead of transmitting 0 over some synthetic bit subchannels, as in the classical Arikan construction, it was suggested to transmit some weighted sum of the symbols transmitted over the other subchannels (dynamic frozen symbols). The particular weighting coefficients are selected in such way, so that the obtained code is a subcode of an extended BCH code. This enables one to obtain codes with substantially higher minimum distance compared to classical polar codes. The codes constructed in this way provide much better performance compared to known LDPC and turbo codes.
V.D. Miloslavskaya has suggested a method for shortening polar (sub)codes, which enables one to obtain codes of arbitrary length, which is not possible in the framework of the classical Arikan construction.
This page presents a database of some polar subcodes. These codes can be decoded using the same techniques as Arikan polar codes.
This construction is used in the 5G channel coding standard.
Sequential decoding of polar codes
V.D. Miloslavskaya and P. V. Trifonov have suggested a sequential decoding algorithm for polar codes. As opposed to the list decoding algorithms suggested by some researchers, the proposed approach avoids a large fraction of useless computation. As a result, the decoding complexity appears to be less, and the performance is better compared to the case of LDPC and turbo codes with similar parameters. This approach can be extended to the case of polar codes with arbitrary kernel, as well as short Reed-Solomon codes. Further complexity reduction techniques for this approach were developed by G. Trofimiuk and N. Iakuba.
Polar codes with large kernels
Polar codes with large kernels provide much better scaling of their performance with code length compared to classical codes with Arikan kernel. However, until recently there were no efficient decoding algorithms for such codes. G.A. Trofimiuk and P.V. Trifonov suggested a novel window-based algorithm, which provides fast decoding of polar codes with kernels of dimension 2^{m}, as well as a family of kernels, for which this algorithm becomes particularly efficient. As a result, polar subcodes with large kernels under list decoding provide better performance compared to similar codes with Arikan kernel with the same decoding complexity. In some cases we obtain both better performance and lower decoding complexity compared to codes with Arikan kernel.
IEEE Communications Society Best Readings in Polar Coding
The following papers of our team were included into IEEE Communications Society Best Readings in Polar Coding:
- P. Trifonov, “Efficient Design and Decoding of Polar Codes,” IEEE Transactions on Communications, vol. 60, no. 11, pp. 3221-3227, November 2012.
- P. Trifonov and V. Miloslavskaya, “Polar Subcodes,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 2, pp. 254-266, February 2016.
- P. Trifonov, “A Score Function for Sequential Decoding of Polar Codes,” in Proc. IEEE International Symposium on Information Theory (ISIT), Vail, USA, June 2018.
- G. Trofimiuk and P. Trifonov, “Efficient Decoding of Polar Codes with Some 16×16 Kernels,” in Proc. IEEE Inf. Theory Workshop (ITW), Guangzhou, China, November 2018.
References
44. |
Window Processing of Binary Polarization Kernels
IEEE Transactions on Communications,
69(7)
July
2021
Keywords: polar |
43. |
Relaxed Decoding of Polar Codes with Large Kernels
IEEE Communications Letters,
25(5):1520-1523
May
2021
Keywords: polar |
42. |
Recursive Trellis Processing of Large Polarization
Kernels
Proceedings of IEEE International Symposium on Information Theory
2021
Keywords: polar |
41. |
Fast Block Sequential Decoding of Polar Codes
IEEE Transactions on Vehicular Technology,
69(10):10988 - 10999
October
2020
Keywords: polar |
40. |
An Approximate Method for Construction of Polar
codes with Kernels over F2t
IEEE Communications Letters,
24(9)
September
2020
Keywords: polar |
39. |
Design of BCH Polarization Kernels with Reduced
Processing Complexity
IEEE Communications Letters,
24(7)
July
2020
Keywords: polar |
38. |
Randomized Polar Subcodes with Optimized
Error Coefficient
IEEE Transactions on Communications,
68(11):6714 - 6722
2020
Keywords: polar |
37. |
Convolutional Polar Kernels
IEEE Transactions on Communications,
68(12):7352 - 7361
2020
Keywords: polar |
36. |
Recursive Trellis Decoding Techniques of Polar Codes
Proceedings of IEEE International Symposium on Information Theory
2020
Keywords: polar |
35. |
On Distance Properties of Convolutional Polar Codes
IEEE Transactions on Communications,
67(7):4585 - 4592
July
2019
ISSN: 1558-0857
Keywords: convolutional polar, b-MERA |
34. |
Successive and Two-Stage Systematic Encoding of Polar Subcodes
IEEE Wireless Communications Letters,
8(3):877 - 880
June
2019
ISSN: 2162-2345
Keywords: polar |
33. |
Компактная спецификация полярных кодов
Информационно-Управляющие Системы,
98(1):40-47
2019
ISSN: 2541-8610
Keywords: polar |
32. |
Polar Subcodes for Encoding and Blind Decoding of Variable-Sized Data Blocks
12th International ITG Conference on
Systems, Communications and Coding
2019
Keywords: polar |
31. |
On Construction of Polar Subcodes with Large Kernels
Proceedings of IEEE International Symposium on Information Theory
2019
Keywords: polar |
30. |
Reduced complexity window processing of binary
polarization kernels
Proceedings of IEEE International Symposium on Information Theory
2019
Keywords: polar |
29. |
A Lower Bound on Minimum Distance of
Convolutional Polar Codes
Proceedings of IEEE International Symposium on Information Theory
2019
Keywords: polar |
28. |
Trellis-based Decoding Techniques for Polar Codes
with Large Kernels
Proceedings of IEEE Information Theory Workshop
2019
Keywords: polar |
27. |
Construction of binary polarization kernels for low complexity window processing
Proceedings of IEEE Information Theory Workshop
2019
Keywords: polar |
26. |
Randomized Chained Polar Subcodes
Proceedings of Wireless Communications and Networking Conference Workshops
, page 25-30.
2018
Keywords: polar |
25. |
Chained Successive Cancellation Decoding of the Extended Golay Code
Proceedings of Iran Workshop on Communication and Information Theory
2018
Keywords: polar |
24. |
A score function for sequential decoding of polar codes
Proceedings of IEEE International Symposium on Information Theory
2018
Keywords: polar |
23. |
Algebraic matching techniques for fast decoding of polar codes with Reed-Solomon kernel
Proceedings of IEEE International Symposium on Information Theory
2018
Keywords: polar |
22. |
Efficient SC Decoding of Convolutional Polar Codes
Proceedings of International Symposium on Information Theory and Applications
2018
Keywords: convolutional polar, b-MERA |
21. |
Efficient decoding of polar codes with some 16 x 16 kernels
Proceedings of IEEE Information Theory Workshop
2018
Keywords: polar |
20. |
Chained polar subcodes
Proceedings of 11th International ITG Conference on Systems, Communications and Coding
2017
Keywords: polar |
19. |
Star polar subcodes
Proceedings of IEEE Wireless Communications and Networking Conference Workshops
2017
Keywords: polar |
18. |
A Randomized Construction of Polar Subcodes
Proceedings of IEEE International Symposium on Information Theory
, page 1863-1867.
2017
Keywords: polar |
17. |
Fast Encoding of Polar Codes with Reed-Solomon Kernel
IEEE Transactions on Communications,
64(7):2746-2753
July
2016
Keywords: polar |
16. |
Polar subcodes
IEEE Journal on Selected Areas in Communications,
34(2):254-266
February
2016
Keywords: polar |
15. |
Hybrid Decoding of Interlinked Generalized Concatenated Codes
Proceedings of 9th International Symposium on Turbo Codes and Iterative Information Processing
2016
Keywords: polar |
14. |
Directed Search Decoding of Polar Codes with Reed-Solomon kernel
Proceedings of XV International Symposium "Problems of Redundancy in Information and Control Systems"
2016
Keywords: polar |
13. |
Shortened Polar Codes
IEEE Transactions on Information Theory,
61(9):4852-4865
September
2015
ISSN: 0018-9448
Keywords: polar |
12. |
Design of Polar Codes for Rayleigh Fading Channel
Proceedings of The International Symposium on
Wireless Communication Systems
2015
Keywords: polar |
11. |
Block Sequential Decoding of Polar Codes
Proceedings of International Symposium on
Wireless Communication Systems
2015
Keywords: polar |
10. |
Multilevel Buckets for Sequential Decoding of Polar Codes
Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Radio Communications
2015
Keywords: polar |
9. |
Sequential Decoding of Polar Codes
IEEE Communications Letters,
18(7):1127 - 1130
July
2014
Keywords: polar |
8. |
Successive Cancellation Decoding
of Reed–Solomon Codes
Problems of Information Transmission,
50(4)
2014
Keywords: polar |
7. |
Binary Successive Cancellation Decoding of Polar Codes with Reed-Solomon Kernel
Proceedings of IEEE International Symposium on Information Theory
, page 2972 - 2976.
2014
Keywords: polar |
6. |
Sequential Decoding of Reed-Solomon Codes
Proceedings of International Symposium on Information Theory and its Applications
, page 466-470.
2014
Keywords: polar |
5. |
Twisted polar codes
Proceedings of International Symposium on Information Theory and Its Applications
, page 456-460.
2014
Keywords: polar |
4. |
Successive Cancellation Permutation Decoding of Reed-Solomon Codes
Proceedings of IEEE Information Theory Workshop
, page 386 - 390.
2014
Keywords: polar |
3. |
Sequential Decoding of Polar Codes with Arbitrary Binary Kernel
Proceedings of IEEE Information Theory Workshop
, page 376 - 380.
2014
Keywords: polar |
2. |
Polar Codes with Dynamic Frozen Symbols and Their Decoding by Directed Search
Proceedings of IEEE Information Theory Workshop
, page 1-5.
2013
Keywords: polar |
1. |
Efficient Design and Decoding of Polar Codes
IEEE Transactions on Communications,
60(11):3221 - 3227
November
2012
Keywords: polar |