The SC-decoder was the first decoder proposed for polar codes by Arikan in [1]. The SC decoder consists of $N$ decision elements (DE) for the $N$ bits of $\hat{\mathrm{u}}_{1}^{N}$. Each of the DE computes the hard decision output based on the observed channel output $y_{1}^{N}$ and the previously decoded bits (e.g. the $k^{th}$ DE would compute $\mathrm{\hat{u}}_{k}$ using $y_{1}^{N}$ and $\mathrm{\hat{u}}_{1}^{k-1})$, and thus the name "Successive".

The DE computes the Likelihood ratio:

$L_{i}\triangleq\dfrac{W_{N}^{(i)}\left(y_{1}^{N},\mathrm{\hat{\mathrm{u}}_{1}^{i-1}\mid0}\right)}{W_{N}^{(i)}\left(y_{1}^{N},\mathrm{\hat{\mathrm{u}}_{1}^{i-1}\mid1}\right)}$

and then the hard decision per DE is generated according to:

$\mathrm{\mathrm{\hat{u}}_{k}}=\begin{cases}0, & \,\,\,\,L_{k}\geq1\\1, & \,\,\,\,\mathrm{L_{k}<1}\end{cases}$

The DEs of indices that belongs to the set $\mathbb{A}^{c}$, sets the $\mathrm{\hat{u}}_{k}=0$ (the frozen bits positions and values could be considered as the decoder's prior knowledge).