Improving successive cancellation decoding of polar codes by usage of inner block codes

Since polar codes are capacity-achieving codes, there have been various research works devoted to devising efficient implementation methods as well as improving error-correction performance. Since quantization is a critical implementation issue, in this paper, a nonuniform quantization method is proposed for the successive cancellation (SC) decoder of polar codes, which finds quantization boundary values based on the analysis of various quantization levels over the additive white Gaussian noise channel. Since low computational complexity, high reliability, and efficient memory management are required in the next-generation communication and memory systems, 2-4 bit precision levels are mainly considered in the proposed nonuniform quantization method. Depending on the presence of erasure, quantization levels are divided into three types, and the message alphabets and update rules are derived for each type. Also, a construction method of polar codes suitable for the proposed nonuniform-quantized SC decoder is proposed, which simultaneously determines the information set and the quantization boundary values based on the density evolution analysis and an upper bound of the block error probability. To determine quantization boundary values, a multivariate objective function is defined and an iterative coarse-to-fine search algorithm to minimize this objective function is proposed. In addition, a scaling method of quantizer output values is proposed when the number of quantization levels of quantizer is smaller than the number of quantization levels of decoder. Finally, simulation results confirm that the proposed nonuniform-quantized SC decoder shows better error-correction performance, lower decoding complexity, and higher memory efficiency compared to the best known uniform-quantized SC decoder.

Under successive cancellation (SC) decoding, polar codes are inferior to other codes of similar blocklength in terms of frame error rate. While more sophisticated decoding algorithms such as list- or stack-decoding partially mitigate this performance loss, they suffer from an increase in complexity. In this paper, we describe a new flavor of the SC decoder, called the SC flip decoder. Our algorithm preserves the low memory requirements of the basic SC decoder and adjusts the required decoding effort to the signal quality. In the waterfall region, its average computational complexity is almost as low as that of the SC decoder.

In recent days, polar codes have attracted the attention of Industry and Academia. It is one of a linear block error correction codes. Polar codes are one of the capacity achieving algorithms over a wide range of channels, making them very attractive from a theoretical perspective. Polar codes are shown to be instances of concatenated codes. It has been shown that the effect of a polar code can be enhanced by showing the multistage decoding algorithm with log-likelihood based Successive List decoding (SCL). However, SCL decoders are no longer optimal in terms of frame error rate. It is believed that the machine learning based algorithm can improve the FER for the decoded bit. Thus, the logistic regression algorithm is proposed in this paper. The proposed method offers an optimized solution to the decoded bits. The simulation result shows that with sufficient training, this method can provide a lower frame error rate for different code lengths.

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , we can compute the weight distribution of any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polar cosets</i> in time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixing factor</i> . To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> = 128.

The neural successive cancellation (NSC) decoder with tanh-based modified log-likelihood ratio (LLR) is proposed for reducing the decoding latency of polar codes over free space optical (FSO) turbulence channel. The conventional successive cancellation (SC) decoder is partitioned into multiple sub-blocks, which are replaced by multiple sub neural network (NN) decoders with tanh-based modified LLR. The recursive characteristic of the polar sequences reliability ranking given in 5G standard enables the sub-NN decoder to be uniquely determined by code length and the number of information bits. Confirmed by the simulation, the bit error rate (BER) performance of NSC decoder with tanh-based modified LLR is close to the conventional SC decoder over turbulence channel for the practical-length polar codes. Regarding turbulence-stability, the NSC decoder trained in moderate and strong turbulence conditions have stable performance in a wide range of turbulence conditions. Moreover, in comparison of decoding latency, the NSC decoder with tanh-based modified LLR takes less than 25% time steps of SC decoder in the same code length.

Polar codes have recently emerged as one of the most favorable capacityachieving error correction codes due to their low encoding and decoding complexity. However, because of the large code length required by practical applications, the few existing successive cancellation (SC) decoder implementations still suffer from not only high hardware cost but also long decoding latency. In this paper, a data-flow graph (DFG) for the SC decoder is derived. A complete hardware architecture is first derived for the conventional tree SC decoder and the feedback part is presented next. Precomputation look-ahead technique is exploited to reduce the achievable minimum decoding latency. Substructure sharing is used to design a merged processing element (PE) for higher hardware utilization. In order to meet throughput requirements for a diverse set of application scenarios, a systematic approach to construct different overlapped SC polar decoder architectures is also presented. Compared with a conventional <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$N$</tex></formula> -bit tree SC decoder, the proposed overlapped architectures can achieve as high as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(N-1)$</tex></formula> times speedup with only <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$({N}\log_{2}{N})/2$</tex></formula> merged PEs. The proposed pre-computation approach leads to a 50% reduction in latency for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${N} &gt; 2^{7}$</tex></formula> , and 40% reduction for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$N \leq 2^{7}$</tex></formula> .

In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automorphisms in the upper-diagonal linear (UTL) group, and thus render SC-based AE decoding effective. Demonstrated by examples, these new polar codes under AE decoding outperform conventional polar codes under SC list decoding in terms of error rate, while keeping the latency comparable to SC decoding. Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.

In this paper, an improvement for successive cancellation (SC) decoding is presented for polar codes. This improvement is based on applying Creeper approach that was used to decode convolutional codes. The proposed decoder combines a stack and a Fano approaches and allows to significantly decrease complexity of Successive Cancellation Fano decoder at the cost of a slight increase in space complexity. We also show how to combine SC-Creeper decoding with bit-flipping approach to further improve the performance of SC-Creeper due to introducing additional decoding attempts.Simulation results provided in this paper show that proposed SC-Creeper and SC-Creeper Flip decoders with carefully selected parameters have the same performance as both Successive Cancellation List decoder with list size 8 and Fano decoder, while having much smaller complexity.

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that "interpolates" between RM and polar codes, call this family C-inter = {C-alpha : alpha is an element of [0, 1]}, where C-alpha vertical bar(alpha=1) is the original polar code, and C-alpha vertical bar(alpha=0) is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of a. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family C-inter even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel.

This paper presents a general decoder architecture for 5G LDPC and polar codes. To achieve the hardware reuse maximization, both LDPC and polar codes are decoded by belief propagation (BP) algorithm on sparse bipartite graphs. However, by counting the number of short cycles on the polar bipartite graphs, we observe that numerous short cycles cause the poor performance than conventional decoding algorithms, like successive cancellation (SC) and BP decoding. To reduce the impact of short cycles, a LDPC-like BP list (BPL) decoding algorithm is proposed for polar codes, which is applied on L different polar bipartite graphs constructed by different permuted polar factor graphs. Finally, the simulation shows that the proposed general decoder for polar codes can obtain beyond SC and BP error performance with lower decoding latency and computational complexity, and even near successive cancellation list (SCL) performance in high signal-noise ratios (SNRs).

By adopting successive cancellation list decoding (SCL), polar codes demonstrate competitive error correction performance over LDPC and Turbo codes. However, SCL decoding suffers from high computational complexity and long decoding latency, especially when the list size is very large. Successive cancellation flip (SCF), as another decoding algorithm that can achieve high error correction performance, has a complexity that is close to that of successive cancellation (SC) decoding. With the observation that SCL and SCF decoding are similar at giving more chances to inspect possible codewords simultaneously or sequentially, a novel hybrid decoder is proposed in this paper, which essentially combines the ideas of SCF and SCL decoders. Moreover, in order to compensate for the degradation of performance caused by the reduction of path splitting and further reduce the decoding latency, the convolutional polar codes are adopted with a designed bit-flipping set. Simulation results demonstrate that the proposed decoder achieves the reduction of decoding latency while attaining better performance than conventional CRC-aided SCL decoder.

Polar codes are error correction codes developed by Erdal Arikan which achieves channel capacity and its reduced complexity makes it more attractive and successful. In the past several decades we are searching for some techniques which will reduce the effect of noise to a very low value so that we can reproduce the transmitted signal more accurately. Erdal Arkan come up with a new concept which known as polar codes, here we use a channel polarization concept and hence named as polar codes. The discovery of polar code is standing as a milestone in coding theory and lot of researches are carrying out on this topic. Variety of schemes have been proposed over these years for the generation, encoding and decoding of polar codes. The important area of research is the decoder section and most widely used one is successive cancellation decoder[3]. Reduced complexity is the most attractive feature with an overall encoding and decoding complexity of O(NlogN) for a block size of N, which leads to its great success. Polar code concept is used in the most promising 5G technology[10]. In this work a complete implementation of the polar Encoder and decoder is discussed with its MATLAB and Verilog implementations

This paper presents the first ASIC implementation of a successive cancellation (SC) decoder for polar codes. The implemented ASIC relies on a semi-parallel architecture where processing resources are reused to achieve good hardware efficiency. A speculative decoding technique is employed to increase the throughput by 25% at the cost of very limited added complexity. The resulting architecture is implemented in a 180nm technology. The fabricated chip can be clocked at 150 MHz and uses 183k gates. It was verified using an FPGA testing setup and provides reference for the true silicon complexity of SC decoders for polar codes.

Successive cancellation list decoder, which is generalized from successive cancellation decoder by increasing list size, is an effective decoding algorithm for polar codes. However, the list length is invariant throughout the decoding process which leads to redundant computation. To avoid this redundant computation, we propose an innovative and heuristic method to adjust the list length according to the variance of path metric values during decoding procedure and signal-to-noise ratio. With decision threshold proportional to the best path, the proposed method fully considers the trend of path metric change. Experimental results demonstrate that our proposed algorithm can reduce computational complexity and improve performance compared with successive cancellation list decoder. For (1024, 512) CRC-aided polar codes, the proposed algorithm can achieve ten times better performance at point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{SNR}=3\mathrm{dB}$</tex> with less computational complexity compared with conventional algorithms.

Typically, forward error correction codes are designed based on the minimization of the error rate for a given code rate. However, for applications that incorporates hybrid automatic repeat request (HARQ) protocol, the throughput is a more important performance metric than the error rate. Polar codes, a new class of error correction codes with simple rate matching and low complexity decoders, can be optimized efficiently for maximization of the throughput. In this paper, we first introduce a method to design throughput-maximizing polar codes for successive cancellation decoding (SCD). Furthermore, since the optimized codes for SCD are not optimal for SC list decoders (SCLD), we propose a rate matching algorithm to find the best rate for the SCLD decoders while using the polar codes optimized for SCD. The resulting codes provide throughput close to capacity with low decoding complexity when used with Type-I HARQ.