Polar code

Polar code[1]

Description

In its basic version, a binary linear polar code encodes \(K\) message bits into \(N=2^n\) bits. The linear transformation that defines the code is given by the matrix \(G^{(n)}=B_N G^{\otimes n}\), where \(B_N\) is a certain \(N\times N\) permutation matrix, and \(G^{\otimes n}\) is the \(n\)th Kronecker power of the \(2\times 2\) kernel matrix \(G=\left[\begin{smallmatrix}1 & 0\\ 1 & 1 \end{smallmatrix}\right]\). To encode \(K\) message bits, one forms an \(N\)-vector \(u\) in which \(K\) coordinates represent the message bits. The remaining \(N-K\) coordinates are set to some fixed values and are said to be frozen. The codeword \(x \in \{0,1\}^N\) is obtained as \(x=u G^{\otimes n}\).

The choice of the frozen coordinates depends on the communication channel, and they correspond to the least reliable bits on the output of the channel under a particular decoding procedure called successive cancellation decoding. If the communication channel is input-symmetric, the values of the frozen bits can be set to zero.

There are multiple variants of the above basic construction, in particular relying on other kernel matrices. The codes can be defined for nonbinary alphabets, and they can be adjusted to support tasks such as lossless and lossy compression, successive refinement, communication over the mulitple access channel, communication over the wiretap channel, and many others.

Protection

Protects against various types of noise in the communication channel, for instance, errors, erasures, or other types of noise. Distance plays no role in the analysis of its properties, and is much lower than the largest possible value given \(K,N\).

Rate

Supports reliable transmission at rates \(K/N\) approaching the Shannon capacity of the channel.

Decoding

Successive cancellation (SC) decoder [1].Successive cancellation list (SCL) decoder [2] and a modification utilizing sequence repetition (SR-List) [3].Soft cancellation (SCAN) decoder [4][5].Belief propagation (BP) decoder [6].

Threshold

Achieves Shannon capacity of the binary-input memoryless channel under successive cancellation decoder [1].

Realizations

Code control channels for the 5G NR (New Radio) interfaces [7].

Notes

For more details, see Ch. 32 of Ref. [8].See Kaiserslautern database [9] for explicit codes.See pretty-good-codes database [10] for explicit representatives and benchmarking.Codes have been benchmarked using AFF3CT toolbox [11].

Parents

Linear binary code
Generalized concatenated code — Polar codes can be represented as generalized concatenations of their kernels.

Cousins

Reed-Muller (RM) code — RM codes rely on the same generator matrix, but place message bits in different coordinates.
Quantum polar code

References

[1]: E. Arikan, “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels”, IEEE Transactions on Information Theory 55, 3051 (2009). DOI
[2]: I. Tal and A. Vardy, “List Decoding of Polar Codes”, IEEE Transactions on Information Theory 61, 2213 (2015). DOI
[3]: Yuqing Ren et al., “A Sequence Repetition Node-Based Successive Cancellation List Decoder for 5G Polar Codes: Algorithm and Implementation”. 2205.08857
[4]: U. U. Fayyaz and J. R. Barry, “Low-Complexity Soft-Output Decoding of Polar Codes”, IEEE Journal on Selected Areas in Communications 32, 958 (2014). DOI
[5]: U. U. Fayyaz and J. R. Barry, “Polar codes for partial response channels”, 2013 IEEE International Conference on Communications (ICC) (2013). DOI
[6]: E. Arkan, “A performance comparison of polar codes and Reed-Muller codes”, IEEE Communications Letters 12, 447 (2008). DOI
[7]: 3rd Generation Partnership Project (3GPP), Technical specification group radio access network, 3GPP TS 38.212 V.15.0.0, 2017.
[8]: W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
[9]: Michael Helmling, Stefan Scholl, Florian Gensheimer, Tobias Dietz, Kira Kraft, Stefan Ruzika, and Norbert Wehn. Database of Channel Codes and ML Simulation Results. URL, 2022.
[10]: G. Liva, F. Steiner. “pretty-good-codes.org: Online library of good channel codes”, URL: http://pretty-good-codes.org/
[11]: A. Cassagne et al., “AFF3CT: A Fast Forward Error Correction Toolbox!”, SoftwareX 10, 100345 (2019). DOI

Cite as:

“Polar code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polar

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/polar.yml.