channel (PUCCH) and the physical uplink shared channel (PUSCH). In the downlink, polar codes are used to encode the downlink control information (DCI) over the physical downlink control channel (PDCCH), and the payload in the physical broadcast channel (PBCH). Table \[tab:sched\_chan\] summarizes the encoding chain parameters and bounds.

In 5G applications, the number of information bits, $A$, is fixed and a codeword of length $E$ is created to achieve the desired rate $R=A/E$ required by upper communication layers. To accommodate polar codes to this requirement, a mother polar code of length $N = 2^n$ is initially constructed, and the desired code length $E$ is matched via puncturing, shortening or repetition. The mother code length $N$ is lower bounded by $N_{min} = 32$, while the value of the upper bound $N_{max}$ depends on the channel used, being $N_{max} = 512$ for downlink and $N_{max} = 1024$ for uplink. An ulterior upper bound is imposed by the minimal accepted code rate of $\frac{1}{8}$.

Figure \[fig:5Gchain\] portrays the set of encoding operations envisaged by the 5G polar codes framework. Vector $\mathbf{a}$ contains the $A$ information bits to be transmitted using a payload of $G$ code bits. Depending on code parameters, the message may 