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Probl. Peredachi Inf., 2018, Volume 54, Issue 4, Pages 3–34 (Mi ppi2278)  

Coding Theory

Polar codes with higher-order memory

H. Afşerab, H. Deliça

a Wireless Communications Laboratory, Department of Electrical and Electronics Engineering, Boğaziçi University, Istanbul, Turkey
b Department of Electrical and Electronics Engineering, Adana Science and Technology University, Adana, Turkey

Abstract: We introduce a construction of a set of code sequences $\{C_n^{(m)}:\: n\ge 1, m\ge 1\}$ with memory order $m$ and code length $N(n)$. $\{C_n^{(m)}\}$ is a generalization of polar codes presented by Arıkan in [1], where the encoder mapping with length $N(n)$ is obtained recursively from the encoder mappings with lengths $N(n-1)$ and $N(n-m)$, and $\{C_n^{(m)}\}$ coincides with the original polar codes when $m = 1$. We show that $\{C_n^{(m)}\}$ achieves the symmetric capacity $I(W)$ of an arbitrary binary-input, discrete-output memoryless channel $W$ for any fixed $m$. We also obtain an upper bound on the probability of block-decoding error $P_e$ of $\{C_n^{(m)}\}$ and show that $P_e=O (2^{-N^\beta})$ is achievable for $\beta<1/[1+m(\phi-1)]$, where $\phi\in (1;2]$ is the largest real root of the polynomial $F(m,\rho)=\rho^m-\rho^{m-1}-1$. The encoding and decoding complexities of $\{C_n^{(m)}\}$ decrease with increasing $m$, which proves the existence of new polar coding schemes that have lower complexity than Arıkan's construction.

Funding Agency Grant Number
Bilimsel Araştırma Projeleri 11A02D10
This work was supported by Boğaziçi University Research Fund under Project 11A02D10. H. Afşer was also supported by Aselsan Elektronik A.Ş.


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English version:
Problems of Information Transmission, 2018, 54:4, 301–328

Bibliographic databases:

UDC: 621.391.15
Received: 17.05.2017
Revised: 06.08.2018
Accepted:07.08.2018

Citation: H. Afşer, H. Deliç, “Polar codes with higher-order memory”, Probl. Peredachi Inf., 54:4 (2018), 3–34; Problems Inform. Transmission, 54:4 (2018), 301–328

Citation in format AMSBIB
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\by H.~Af{\c s}er, H.~Deli{\c c}
\paper Polar codes with higher-order memory
\jour Probl. Peredachi Inf.
\yr 2018
\vol 54
\issue 4
\pages 3--34
\mathnet{http://mi.mathnet.ru/ppi2278}
\transl
\jour Problems Inform. Transmission
\yr 2018
\vol 54
\issue 4
\pages 301--328
\crossref{https://doi.org/10.1134/S0032946018040014}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060767898}


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