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International Symposium on Turbo Codes and Iterative Information Processing

On the Marginalization of Polarizing Kernels

Valerio Bioglio, Ingmar Land

Published Dec 1, 2018• 25 citations
Publisher

Abstract

In this paper, we analyze the decoding complexity of polarizing kernels of size larger than p = 2, proposing a general procedure which can be used to marginalize kernels of any size. We use this method to evaluate the decoding equations for optimal binary kernels up to size p = 8. Finally, we experimentally show that, even if T2and T8share the same polarization exponent, their performance may vary under list decoding, making large kernels of interest for long polar codes.

DOI
10.1109/istc.2018.8625378

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In this paper, we analyze the decoding complexity of polarizing kernels of size larger than p = 2, proposing a general procedure which can be used to marginalize kernels of any size. We use this method to evaluate the decoding equations for optimal binary kernels up to size p = 8. Finally, we experimentally show that, even if T2and T8share the same polarization exponent, their performance may vary under list decoding, making large kernels of interest for long polar codes.
Polar codes constructed with large polarization kernels were recently proven to be able to achieve optimal finite-length scaling properties. However, straightforward decoding for large kernel polar codes introduces a complexity coefficient that is exponential to the kernel sizes, which makes such codes generally believed to be impractical. In this paper, we present a new method that decodes large kernel polar codes with a complexity coefficient that is polynomial to the kernel sizes. This could facilitate the implementation of large kernel polar codes for practical use in the future.Successive cancellation decoding for large kernel polar codes requires calculation on the probabilities of its bit channels. Similar to conventional polar codes, those bit channels follow a recursive relation, which make this calculation boil down to computing the probabilities for bit channels of a single polarization kernel. This kernel-level computation can be shown equivalent to soft-output maximum-likelihood (ML) decoding on a single bit of a linear block code. In our proposed method, we first use linear operations to represent the considered linear block code as a polar code with dynamically frozen bits, and then use a modified polar list decoder to get an approximate value on the soft-output of the desired bit. This method is motivated by the observation that at short block lengths, polar list decoding with a large enough list size can well-approximate ML decoding. The proposed low-complexity method allows us to decode polar codes constructed with a 64 × 64 polarization kernel with scaling exponent µ ≈ 2.87 for the first time.
Window decoding is useful for decoding polar codes defined by kernels that do not have Arıkan's original form. We modify arbitrary polarization kernels of size 2t × 2t to reduce the time complexity of window decoding. This modification is based on the permutation of the columns of the kernel. This method is applied to some of the kernels constructed in the literature of size 16 and 32, with different error exponents and scaling exponents such as eNBCH kernel. It is shown that this method reduces the complexity of the window decoding significantly without affecting the performance.
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Polar codes with large kernels offer improved error-correction performance but suffer from high decoding complexity due to costly marginalization operations. In this work, we propose neural network (NN)-based approximations of these marginalization functions, enabling efficient decoding for arbitrary linear kernels. We show that training on all-zero partial codewords is sufficient for linear kernels, significantly reducing the dataset size and allowing for smaller, faster NN models. Our method is evaluated within a Successive Cancellation (SC) decoder, being compatible with any blocklength, code rate, or SNR. Experimental results for kernel size 16 demonstrate strong performance and substantial complexity savings over exact marginalization.
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.