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Open AccessArticle

The Fractality of Polar and Reed–Muller Codes

Signal Processing and Speech Communication Laboratory, Graz University of Technology, 8010 Graz, Austria
This paper is an extended version of our paper published in the 2015 NEWCOM# Emerging Topics in Modulation and CodingWorkshop and in the 2016 International Zürich Seminar on Communications, Zurich, Switzerland, 2–4 March 2016.
Entropy 2018, 20(1), 70; https://doi.org/10.3390/e20010070
Received: 16 October 2017 / Revised: 10 January 2018 / Accepted: 15 January 2018 / Published: 17 January 2018
(This article belongs to the Section Information Theory, Probability and Statistics)
The generator matrices of polar codes and Reed–Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed–Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals. View Full-Text
Keywords: polar codes; Reed–Muller codes; fractals; self-similarity polar codes; Reed–Muller codes; fractals; self-similarity
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Geiger, B.C. The Fractality of Polar and Reed–Muller Codes. Entropy 2018, 20, 70.

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