Polar Coding Notes IV

Consider a $G_N$-coset code with parameter $(N, K, \cal A , u_{\cal A^c})$. Let $u_1^N$ be encoded into a codeword $x_1^N$, let $x_1^N$ be sent over the channel $W^N$, and let a channel output $y_1^N$ be received. The decoder’s task is to generate an estimate $\hat u_1^N$ of $u_1^N$, given knowledge of $\cal A$, $u_{\cal A^c}$, and $y_1^N^[1]$.

Successive Cancellation Decoder

Individual $G_N$-coset codes will be identified by a parameter vector $(N, K, \cal A, u_{\cal A^c})$, where $K$ is the code dimension and specifies the size of $\cal A$. The ratio $K/N$ is called the code rate. We will refer to $\cal A$ as the information set and to $u_{\cal A^c} \in {\cal X}^{N−K}$ as frozen bits or vector.
We obtain a mapping from source blocks $u_{\cal A}$ to codeword blocks $x_1^N$ as

$$x_1^N = u_1^N G_N = u_{\cal A}G_N(\cal A) \oplus u_{\cal A^c}G_N(\cal A^c)$$

Reference:

1. E. Arikan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Trans. on Information Theory, vol.55, no.7, pp.3051–3073, July 2009.
2. A. D. Wyner and J. Ziv. A theorem on the entropy of certain binary sequences and applications (Part I). IEEE Trans.Inform.Theory, vol.19, no.6, pp.769-772, Nov.1973.
3. Abbas El Gamal and Young-Han Kim. Network Information Theory. Cambridge University Press. 2011.
4. M.Alsan and E.Telatar. A simple proof of polarization and polarization for non-stationary memoryless channels. IEEE Trans.Info.Theory, vol.62, no.9,pp.4873-4878. 2016.
5. Vincent Y. F. Tan. EE5139R: Information Theory for Communication Systems:2016/7, Semester 1. https://www.ece.nus.edu.sg/
6. Eren Sasoglu. Polarization and Polar Codes. Foundations and Trends in Communications and Information Theory Vol. 8, No. 4 (2011) 259–381

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