Polar code[1]

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.