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IEEE International Symposium on Information Theory (ISIT)

Generalized Concatenated Polarization Kernels

Peter Trifonov

Published Jul 7, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1109/isit57864.2024.10619247

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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.
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.
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.
A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.
Polar coding has found its way into many realms in communications and information theory. In most implementation setups, they are accompanied with the list successive cancellation (LSC) decoding algorithm which is shown to provide a superior error performance compared to the original successive cancellation (SC) decoding method. While the SC decoding is fairly well-studied, the exact math behind LSC's superior performance still remains to be of mystery. Multiple techniques have been proposed to further improve the LSC's error performance or to reduce its computational complexity, which are usually motivated by heuristic reasons and shown through numerical simulations. Most notable example is the CRC-aided LSC, which drastically improves the LSC's performance by concatenating the polar code with some high-rate cyclic redundancy check (CRC) codes.In this paper, we present polar codes that are concatenated with an underlying high-rate convolutional code, which are shown to have superior performances over CRC-aided LSC. We also present a computationally-efficient decoding algorithm for these codes which resembles the techniques used in the Viterbi algorithm, and hence is called the convolutional decoding algorithm. To do this, we revisit the error analysis of the original SC decoding along with the concept of Arıkan's helper genie. We address some shortcomings of the CRC-aided LSC and discuss how to turn around them by emulating a convolutional code instead of a CRC code. Contrary to CRC codes, most of the convolutional codes are not a proper choice for concatenation with polar codes. We introduce the bucketing algorithm to construct suitable punctured convolutional codes for this purpose. The proposed framework can accommodate any such underlying convolutional code, which allows one to search for the optimal convolutional code based on their design parameters.