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arXiv:1209.4818

Recursive Descriptions of Decoding Algorithms and Hardware Architectures for Polar Codes

Noam Presman, Simon N. Litsyn

Published Sep 21, 2012• 3 citations
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Abstract

Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. This description allows an easy development of the first two algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arikan's standard polar codes and for arbitrary polarizing kernels.

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Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. This description allows an easy development of the first two algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arikan's standard polar codes and for arbitrary polarizing kernels.
A novel class of polarization kernels is proposed, which is based on the construction of generalized concatenated codes. The complexity of the kernel processing operation, which is the computational core of the (list) successive cancellation decoder, for the obtained kernels of size 64 is several orders of magnitude less compared to BCH and Yao-Fazeli-Vardy kernels. Polar subcodes based on the obtained kernels under list successive cancellation decoder provide both performance and complexity improvement with respect to those based on the Arikan kernel.
Arikan's recursive code construction is designed to polarize a collection of memoryless channels into a set of good and a set of bad channels, and it can be efficiently decoded using successive cancellation. It was recently shown that the same construction also polarizes channels with memory, and a generalization of successive cancellation decoder was proposed with a complexity that scales like the third power of the channel's memory size. In another line of work, the polar code construction was extended by replacing the block polarization kernel by a convoluted kernel. Here, we present an efficient decoding algorithm for finite-state memory channels that can be applied to polar codes and convolutional polar codes. This generalization is most effectively described using the tensor network formalism, and the manuscript presents a self-contained description of the required basic concepts. We use numerical simulations to study the performance of these algorithms for practically relevant code sizes and find that the convolutional structure outperforms the standard polar codes on a variety of channels with memory.
This brief presents a hardware architecture and algorithmic improvements for list successive cancellation (SC) decoding of polar codes. More specifically, we show how to completely avoid copying of the likelihoods, which is algorithmically the most cumbersome part of list SC decoding. The hardware architecture was synthesized for a block length of N=1024 bits and list sizes L=2, 4 using a UMC 90 nm VLSI technology. The resulting decoder can achieve a coded throughput of 181 Mb/s at a frequency of 459 MHz.
Construction and Decoding Algorithms for for Polar Codes based on 2×22\times22×2 Non-Binary Kernels
Polar codes based on 2×22\times22×2 non-binary kernels are discussed in this work. The kernel over GF(q)\text{GF}(q)GF(q) is selected by maximizing the polarization effect and using Monte-Carlo simulation. Belief propagation (BP) and successive cancellation (SC) based decoding algorithms are extended to non-binary codes. Additionally, a successive cancellation list (SCL) decoding with a pruned tree is proposed. Simulation results show that the proposed decoder performs very close to a conventional SCL decoder with significantly lower complexity.