zeros.

Update rules $f$ and $g$ for left and right child nodes respectively depend on the channel model. In the case of BEC, soft values belong to the set $\{0,1,e\}$, with $e$ representing an erasure, while hard decisions can only take values $0$ and $1$. Update rules are $f(\alpha_1,\alpha_2) = \alpha_1 \oplus \alpha_2$ and $g(\alpha_1,\alpha_2,\beta) = \alpha_2 \lor (\alpha_1 \oplus \beta)$; the erasure algebra imposes that $e \oplus \bullet = e$ and $e \lor \bullet = \bullet$. Continuing with the example of Figure \[figG8\], Figure \[figSCDec\] shows the flux of soft and hard messages passed along the decoding tree if soft vector $\mathbf{y}=[e,0,e,0,e,1,0,1]$ has been received, allowing to correctly decode the message.

Advanced design {#sec:adv}
===============

In this section, we introduce more advanced polar codeing principles. The discussion is not limited to frozen set calculation and encoding/decoding algorithms including techniques for rate matching and assistant bits design for 5G applications.

Frozen set design
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In general, the reliability order of the bit-channels depends on the channel condition and on the code length, and therefore is not universal. This non-universality poses huge practical problems in the construction of polar codes when a large range of code lengths and rates is envisaged. 