# 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

## Rate

## Decoding

## Threshold

## Realizations

## Notes

## 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

## Zoo code information

## 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]
- 3rd Generation Partnership Project (3GPP), Technical specification group radio access network, 3GPP TS 38.212 V.15.0.0, 2017.
- [5]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). 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.