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Tsinghua Science & Technology

Sphere Decoding for Binary Polar Codes with the Modified Multiplicative Repetition Construction

Haiqiang Chen, Yuanbo Liu, Shuping Dang, Qingnian Li, Youming Sun, Xiangcheng Li

Published Dec 30, 2024• 1 citation
Publisher

Abstract

Compared to the successive cancellation (SC)-based decoding algorithms, the sphere decoding (SD) algorithm can achieve better performance with reduced computational complexity, especially for short polar codes. In this paper, we propose a new method to construct the binary polar codes with the modified multiplicative repetition (MR)-based matrix. Different from the original construction, we first design a$2\times 2\ q\text{-ary}$kernel to guarantee the single-level polarization effect. Then, by replacing the new-designed binary companion matrix, a novel strategy is further developed to enhance the polarization in the bit level, resulting in a better distance property. Finally, the SD-based Monte-Carlo (SDMC) method is used to construct MR-based binary polar codes, while the resulting codes without the butterfly pattern are decoded by the SD algorithm. Simulation results show that the proposed method with the SD algorithm can achieve a maximum performance gain of 0.27 dB compared to the original method with slightly lower complexity.

DOI
10.26599/tst.2024.9010030

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Compared to the successive cancellation (SC)-based decoding algorithms, the sphere decoding (SD) algorithm can achieve better performance with reduced computational complexity, especially for short polar codes. In this paper, we propose a new method to construct the binary polar codes with the modified multiplicative repetition (MR)-based matrix. Different from the original construction, we first design a2×2 q-ary2\times 2\ q\text{-ary}2×2 q-arykernel to guarantee the single-level polarization effect. Then, by replacing the new-designed binary companion matrix, a novel strategy is further developed to enhance the polarization in the bit level, resulting in a better distance property. Finally, the SD-based Monte-Carlo (SDMC) method is used to construct MR-based binary polar codes, while the resulting codes without the butterfly pattern are decoded by the SD algorithm. Simulation results show that the proposed method with the SD algorithm can achieve a maximum performance gain of 0.27 dB compared to the original method with slightly lower complexity.
In this paper, we discuss non-binary polar codes using a 2×2 matrix over a Galois field GF(2q) as the kernel. Conventional construction of non-binary polar codes divides the synthesized channels into frozen channels and information channels. Each information channel carries one symbol, i.e., q bits. However, there are many middle channels with insufficient polarization, which cannot carry one symbol of q bits but only i bits, 1≤i<q,i∈Z, at finite block length. In this paper, we consider bit-level construction for multiplicative repetition (MR)-based non-binary polar codes and propose a bit-level construction based on the two following methods. We first calculate the error probability and channel capacity lower bound of each synthesized channel based on the channel degradation method, and then determine both the number and index of the carried bits for each synthesized channel according to the symbol error probability and capacity. To reduce complexity, we also introduce a Monte-Carlo method. We compute the error probability of each synthesized channel carrying i information bits and select the optimal construction that can minimize the union bound of the error probability. Finally, an improved construction-based probabilistic shaping method for MR-based non-binary polar codes is considered. Simulation results show that the proposed construction significantly improved the decoding performance compared with the conventional construction scheme.
Construction and Decoding Algorithms for for Polar Codes based on 2×22\times22×2 Non-Binary Kernels
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Conventional construction of non-binary polar codes divides the synthesized channels into the frozen channels and information channels. Each information channel carries one symbol i.e.qqqbits. However, there are many middle channels with insufficient polarization, which can not carry one symbol ofqqqbits but onlyiiibits,1≤i<q,i∈Z1 \leq i<q, i \in \mathbb{Z}1≤i<q,i∈Z, at finite block lengths. In this paper, we consider the bit-level construction for multiplicative repetition (MR) based non-binary polar codes and propose bitlevel improved construction for channel degradation method and Monte-Carlo method. Simulation results show that the proposed construction can significantly improve the decoding performance compared with the conventional construction scheme.
Polar codes represent an emerging correcting codes with power to approach the capacity of a discrete memoryless channel.However, the decoding performance of polar codes with traditional successive cancellation (SC) algorithm cannot match that of the low-density parity-check or Turbo codes.In this paper, a modified successive cancellation stack (MSCS) polar decoder is proposed.Unlike the original SC stack polar (SCS) decoders, we propose a modified path that can significantly reduce latency of SCS decoders.Simulation results in the binary-input additive white Gaussian noise channel show that modified algorithm has the slightly performance as the SCS algorithm.Moreover, the time and space complexity of the MSCS decoder is much lower than that of the SCS and can be very close to that of the SC in the high SNR regime.
Multi-kernel polar codes, which are constructed by different sizes of kernels make the traditional polar codes have more flexible code length. In this paper, we extend the multi-kernel polar codes to 5×55 \times 55×5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5×55 \times 55×5 kernel, which has the maximum polarization rate. The first row of this kernel is all-one vector, which enable us to seek the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to this kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find this REP node modification can be generalized when multi-kernel polar codes constructed by kernels, which the first row is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for this kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (GREP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. The G-PC and G-REP nodes can further accelerate the decoding speed. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68. 4% reduced latency compared with the traditional SC decoder in all cases considered where frame lengths are (80,135,400,67580,135,400,67580,135,400,675). Index Terms-Multi-kernel polar codes, node modification, fast decoding, 5×55 \times 55×5 kernel matrix
Multi-kernel polar codes, which are constructed by different sizes of kernels make the length of the traditional polar code more flexible. In this paper, we extend the multi-kernel polar codes to 5×5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5×5 kernel with the maximum polarization rate. The first row of this kernel is an all-one vector, which enable us to find the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to the new 5×5 kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find our REP node modification can be generalized to multi-kernel polar codes when the first row of the used kernels is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for the new kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (GREP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68.4% latency reduction compared with the traditional SC decoder in all cases considered where code lengths are 80, 135, 400, and 675.