A Sparse Neural Network Decoder for Non-binary Polar Codes

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

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).

Polar codes can provably achieve the capacity of a symmetric binary discrete memoryless channel and have low encoding and decoding complexity. However, the error rate performance of polar codes decoding in short and moderate length is not very well, moreover, the encoding and decoding of polar codes with the conventional serial mode will lead to poor throughput. In this paper, we propose a hardware architecture of parallel encoding and decoding scheme for polar codes concatenation with LDPC, and take advantage of the parallelism of belief propagation (BP) decoding algorithm of the two codes to reduce the decoding delay. We compare the performance of concatenated scheme with polar codes and investigate the throughput implemented on graphic processing unit (GPU) for Gaussian channel. Experiment results show that the performance of the concatenated scheme outperform only polar codes, and the throughput of the proposed parallel architecture is obviously faster than that of the serial.

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.

Belief propagation (BP) polar code decoder is well-studied from many aspects. This study proposes a hardware optimization to improve performance of polar BP decoder by modifying both processing element (PE) and early stopping criterion (ESC). PE is optimized by using high-speed parallel-prefix Ling adder instead of carry ripple adder and WIB ESC introduced in literature is optimized by removing unnecessary adder array. FPGA implementation is made to verify the idea behind this study. The results show that throughput of polar code belief propagation decoder can be increase significantly by using proposed approach. More accurate comparison can be made by application specific chip design with same parameters. This part of the study will be held in future.

The state-of-the-art soft-output decoder for polar codes is a message-passing algorithm based on belief propagation, which performs well at the cost of high processing and storage requirements. In this paper, we propose a low-complexity alternative for soft-output decoding of polar codes that offers better performance but with significantly reduced processing and storage requirements. In particular we show that the complexity of the proposed decoder is only 4% of the total complexity of the belief propagation decoder for a rate one-half polar code of dimension 4096 in the dicode channel, while achieving comparable error-rate performance. Furthermore, we show that the proposed decoder requires about 39% of the memory required by the belief propagation decoder for a block length of 32768.

In this paper, a novel stopping criterion for Belief Propagation (BP) decoding of polar codes is proposed. The proposed stopping criterion is based on the Bit Difference Ratio (BDR). The BDR stopping method is implemented by monitoring the fluctuation of BDR along with the number of iterations, to decide if the BP decoding process can be terminated. Simulation results indicate that the proposed criterion can provide superiority in iteration reduction compared with the current stopping criteria. Hence, the proposed BDR stopping criterion can effectively reduce the energy waste and decoding latency in systems. Moreover, by loosing the constraint, the proposed stopping criterion can achieve more iteration reduction at the cost of Bit Error Rate (BER) degradation.

Given a bipartite graph, the maximum balanced biclique (MBB) problem, discovering a mutually connected while disjoint sets of equal size with the maximum cardinality, plays a significant role for mining the bipartite graph and has numerous applications. Despite the NP-hardness of the MBB problem, in this paper, we show that an exact MBB can be discovered extremely fast in bipartite graphs for real applications. We propose two exact algorithms dedicated for small dense and large sparse bipartite graphs respectively. For dense bipartite graphs, an O*(1.3803n) algorithm is proposed. This algorithm in fact can find an MBB very fast for small dense bipartite graphs that are common for applications such as VLSI design. This is because, using our proposed novel techniques, the search can fast converge to sufficiently dense bipartite graphs which we prove to be polynomial-time solvable. For large sparse bipartite graphs typical for applications such as biological data analysis, an O*(1.3803 δ) algorithm is proposed, where δ is only a few hundred for large sparse bipartite graphs with millions of vertices. The indispensible optimization that leads to this time complexity is: we transform a large sparse bipartite graph into a limited number of dense subgraphs such that each of the dense subgraphs has up to δ vertices and then apply our proposed algorithm for dense bipartite graphs on each of the subgraphs. To further speed up this algorithm, tighter upper bounds, faster heuristics and more effective reductions are proposed, allowing an MBB to be discovered within a few seconds for bipartite graphs with millions of vertices. Extensive experiments are conducted on synthetic and real large bipartite graphs to demonstrate the efficiency and effectiveness of our proposed algorithms and techniques.

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

This work presents an integrated framework that combines polar coding, spatial modulation, and symbol‐level guessing random additive noise decoding (symbol‐level GRAND) aided detection to enhance the reliability and latency performance of CRC‐aided polar codes in 6G ultrareliable low‐latency communication (URLLC) scenarios. The proposed polar‐coded spatial modulation (PCSM) scheme utilizes (CA‐polar) codes to encode information bits transmitted as antenna index and symbol bits over a MIMO Rayleigh fading channel. The receiver employs a two‐stage detection scheme comprising initial spatial signal processing for antenna and symbol detection, followed by symbol refinement through symbol‐level GRAND with final error correction achieved with CRC‐aided successive cancellation list (CA‐SCL) decoding. A sequential detection and decoding strategy is adopted, in which symbol‐level GRAND refines the detected bits before CA‐SCL polar decoding, reducing error propagation into the polar decoder and thereby lowering the overall decoding complexity. Performance of the scheme is evaluated for polar codeword length of 128 bits for modulation orders of 16‐QAM and 64‐QAM, highlighting the scheme's parametric flexibility under short codeword constraints. Simulations carried out in MATLAB demonstrate that error correction with symbol‐level GRAND‐aided detection achieves an improved bit error rate (BER) and a lower system complexity facilitating real‐time hardware realization.

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.

We propose a new family of polar codes to realize high coding gain, low complexity, and high throughput by introducing a protograph-based design. Our proposed technique, called quasi-cyclic (QC) polar codes, can be highly parallelized without sacrificing decoding complexity. We analyze short cycles in the protograph polar codes and develop a design method to increase the girth. Our approach can resolve the long-standing unsolved problem that belief propagation (BP) decoding does not work well for polar codes due to the inherently short cycles. We demonstrate that a high lifting factor of QC polar codes can improve the performance and that QC polar codes with BP decoding can outperform conventional polar codes with state-of-the-art list decoding. Moreover, we show that a greedy pruning method can improve the performance-complexity trade-off.

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

Motivated by the capacity-approaching property of both binary polar codes (PCs) and the multi-level coding (MLC) scheme, we consider the design of modulated PC over additive white Gaussian noise (AWGN) channels. The capability of modulated PC essentially depends on the selection of information bits set from all the virtual bit channels formed by both modulation and channel polarization. To this end, we characterize the a priori distribution density as the sufficient statistic for the bit channel of individual modulation levels formed by an MLC scheme, in which the codeword of symbols is split into parallel bits streams. According to the factor graph generated by the binary PC, we apply the density evolution (DE) for each binary modulation level. Consequently, distribution densities of every bit within the modulated codeword can be obtained and we select the most reliable bits as information bits and set the complement as frozen bits. Comprehensive simulations demonstrate the validness of this approach compared to the benchmark scheme.

In this chapter, we convert the signal detection in a C-RAN to an inference problem over a bipartite random geometric graph. By passing messages among neighboring nodes, message passing (a.k.a. belief propagation) provides an efficient way to solve the inference problem over a sparse graph. However, the traditional message-passing algorithm does not guarantee to converge, because the corresponding bipartite random geometric graph is locally dense and contains many short loops. As a major contribution of this chapter, we propose a randomized Gaussian message passing (RGMP) algorithm to improve the convergence. The proposed RGMP algorithm demonstrates significantly better convergence performance than the conventional message passing algorithms. In addition, we generalize the RGMP algorithm to a blockwise RGMP (B-RGMP) algorithm, which allows parallel implementation. The average computation time of B-RGMP remains constant when the network size increases.