(perfectly noiseless channels), resulting in channels of extreme capacities. Polar codes provide a low-complexity scheme to construct polarized channels, where the fraction of noiseless channels tends to the capacity of the original B-DMC.

The mathematical foundations of polar codes lie in the discovery of the polarization phenomenon of matrix $\mathbf{G_2} = \left[\begin{smallmatrix} 1&0\\1&1 \end{smallmatrix}\right]$, also known as *basic polarization kernel*. This matrix enables the encoding of a two-bits input vector $\mathbf{u} = [u_0,u_1]$ into code word $\mathbf{d} = [d_0,d_1]$ as $\mathbf{d} = \mathbf{u} \cdot \mathbf{G_2}$, namely having $d_0 = u_0 \oplus u_1$ and $d_1 = u_1$. The polarization effect brought by $\mathbf{G_2}$ is more evident for binary erasure channels (BECs), where the transmitted bit is either received correctly or lost with probability $\delta$. If the input bits are decoded sequentially, bit $u_0$ cannot be recovered as $u_0 = d_0 \oplus d_1$ if any of the code bits has not been received, hence with probability $\delta_0 = (2 - \delta)\delta > \delta$. After $u_0$ has been correctly decoded, input bit $u_1$ can be decoded either as $u_1 = d_1$ or $u_1 = u_0 \oplus d_0$, hence it is sufficient that at least one of the two code bits has been received: 