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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