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IEEE Access

A Fast Two-Phase Monte Carlo Method for Constructing Polar Codes With Arbitrary Binary Kernel

Shiyi Zhang, Zhiliang Huang, Geng Chen, Ming Chen

Published Jan 1, 2019• 4 citations
Publisher

Abstract

We propose a two-phase Monte Carlo (TPMC) method to accelerate the original Monte Carlo (MC) method for constructing polar codes with high-dimensional kernels. In the TPMC method, some of the most reliable and unreliable bits are obtained by Gaussian approximate-density evolution (GA-DE) method in the first phase; in the second phase, these most reliable and unreliable bits are viewed as frozen bits. Then, the MC method are used to evaluate the remaining bits and select some best bits from the remaining bits. Finally, these best bits and the most reliable bits are combined as the information bits of the constructed polar code. By our investigation, most bits can be fixed as frozen bits in the second phase without error performance loss of the constructed polar codes. Because computation of frozen bits can be saved and computation of the GA-DE method can be ignored in contrast to that of the MC method, the TPMC method substantially reduces the complexity of the MC method. Simulation results show that 1) For a$G_{15}^{\otimes 3}$polar code with block length 3375 and code rate 1/2, the TPMC method can fix 3200 bits as frozen bits in the second phase without error performance loss, which reduces the computation cost by approximately 92.6% over the MC method; 2) Polar codes with high-dimensional kernels constructed by the TPMC method outperforms polar codes with the$G_{2}$kernel constructed by the Tal-Vardy method in terms of error performance; 3) With the same computational cost, the TPMC method can construct better polar codes than the MC method.

DOI
10.1109/access.2019.2940544

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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.$q$bits. However, there are many middle channels with insufficient polarization, which can not carry one symbol of$q$bits but only$i$bits,$1 \leq i<q, i \in \mathbb{Z}$, 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.

IEEE Information Theory Workshop (ITW)•Nov 24, 2024•1 citation
We propose a two-phase Monte Carlo (TPMC) method to accelerate the original Monte Carlo (MC) method for constructing polar codes with high-dimensional kernels. In the TPMC method, some of the most reliable and unreliable bits are obtained by Gaussian approximate-density evolution (GA-DE) method in the first phase; in the second phase, these most reliable and unreliable bits are viewed as frozen bits. Then, the MC method are used to evaluate the remaining bits and select some best bits from the remaining bits. Finally, these best bits and the most reliable bits are combined as the information bits of the constructed polar code. By our investigation, most bits can be fixed as frozen bits in the second phase without error performance loss of the constructed polar codes. Because computation of frozen bits can be saved and computation of the GA-DE method can be ignored in contrast to that of the MC method, the TPMC method substantially reduces the complexity of the MC method. Simulation results show that 1) For aG15⊗3G_{15}^{\otimes 3}G15⊗3​polar code with block length 3375 and code rate 1/2, the TPMC method can fix 3200 bits as frozen bits in the second phase without error performance loss, which reduces the computation cost by approximately 92.6% over the MC method; 2) Polar codes with high-dimensional kernels constructed by the TPMC method outperforms polar codes with theG2G_{2}G2​kernel constructed by the Tal-Vardy method in terms of error performance; 3) With the same computational cost, the TPMC method can construct better polar codes than the MC method.
In order to construct high-dimensional core matrix polar codes quickly,a two-stage polar code construction method TPMC based on Monte Carlo(MC) is proposed.In the first stage,the most reliable and unreliable bits are obtained by using the Gauss approximation method with linear complexity.In the second stage,the above bits are fixed as frozen bits and MC method is implemented to measure the error rate of the remaining bit channels.The bits with a lower error rate are selected from the remaining bits,and are combined with most reliable information bits in the first stage to form an information bits set.Simulation results show that compared with MC method,TPMC method can reduce the computational complexity and improve the decoding efficiency.
In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.
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.
Polar coding gives rise to the first explicit family of codes that provably achieve capacity for a wide range of channels with efficient encoding and decoding. But how fast can polar coding approach capacity as a function of the code length? In finite-length analysis, the scaling between code length and the gap to capacity is usually measured in terms of the scaling exponent μ. It is well known that the optimal scaling exponent, achieved by random binary codes, is μ = 2. It is also well known that the scaling exponent of conventional polar codes on the binary erasure channel (BEC) is μ = 3.627, which falls far short of the optimal value. On the other hand, it was recently shown that polar codes derived from ℓ × ℓ binary polarization kernels approach the optimal scaling exponent μ = 2 on the BEC as ℓ→∞, with high probability over a random choice of the kernel. Herein, we focus on explicit constructions of ℓ×ℓ binary kernels with small scaling exponent for ℓ ≤ 64. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent μℓtransforms an underlying BEC into ℓ bit-channels W1, W2>,..., Wℓ. The erasure probabilities of W1, W2>,..., Wℓ, known as the polarization behavior of Kℓ, determine the resulting scaling exponent μ(Kℓ). We first introduce a class of self-dual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing Kℓto that of producing a certain nested chain of only ℓ/2 self-orthogonal codes. We use nested cyclic codes, whose distance is as high as possible subject to the orthogonality constraint, to construct the kernels K32and K64. In order to evaluate the polarization behavior of K32and K64, two alternative trellis representations (which may be of independent interest) are proposed. Using the resulting trellises, we show that μ(K32) = 3.122 and explicitly compute over half of the polarization-behavior coefficients for K64, at which point the complexity becomes prohibitive. To complete the computation, we introduce a Monte-Carlo interpolation method, which produces the estimate μ(K64) ≃ 2.87. We augment this estimate with a rigorous proof that μ(K64) <; 2.97.
We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms stateof the-art constructions using puncturing methods.
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.