Relaxed Polar Codes

Polar codes are the latest breakthrough in coding theory, as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of discrete memoryless channels. Ar{\i}kan's polar encoder and successive cancellation decoder have complexities of $N \log N$, for code length $N$. Although, the complexity bound of $N \log N$ is asymptotically favorable, we report in this work methods to further reduce the encoding and decoding complexities of polar coding. The crux is to relax the polarization of certain bit-channels without performance degradation. We consider schemes for relaxing the polarization of both \emph{very good} and \emph{very bad} bit-channels, in the process of channel polarization. Relaxed polar codes are proved to preserve the capacity achieving property of polar codes. Analytical bounds on the asymptotic and finite-length complexity reduction attainable by relaxed polarization are derived. For binary erasure channels, we show that the computation complexity can be reduced by a factor of 6, while preserving the rate and error performance. We also show that relaxed polar codes can be decoded with significantly reduced latency. For AWGN channels with medium code lengths, we show that relaxed polar codes can have lower error probabilities than conventional polar codes, while having reduced encoding and decoding computation complexities.Comment: Conference version,Relaxed Channel Polarization for Reduced Complexity Polar Coding, accepted for presentation at IEEE Wireless Communications and Networking Conference WCNC 201

Reduced Complexity Belief Propagation Decoders for Polar Codes

Polar codes are newly discovered capacity-achieving codes, which have attracted lots of research efforts. Polar codes can be efficiently decoded by the low-complexity successive cancelation (SC) algorithm and the SC list (SCL) decoding algorithm. The belief propagation (BP) decoding algorithm not only is an alternative to the SC and SCL decoders, but also provides soft outputs that are necessary for joint detection and decoding. Both the BP decoder and the soft cancelation (SCAN) decoder were proposed for polar codes to output soft information about the coded bits. In this paper, first a belief propagation decoding algorithm, called reduced complexity soft cancelation (RCSC) decoding algorithm, is proposed. Let $N$ denote the block length. Our RCSC decoding algorithm needs to store only $5N-3$ log-likelihood ratios (LLRs), significantly less than $4N-2+\frac{N\log_2N}{2}$ and $N(\log_2N+1)$ LLRs needed by the BP and SCAN decoders, respectively, when $N\geqslant 64$. Besides, compared to the SCAN decoding algorithm, our RCSC decoding algorithm eliminates unnecessary additions over the real field. Then the simplified SC (SSC) principle is applied to our RCSC decoding algorithm, and the resulting SSC-aided RCSC (S-RCSC) decoding algorithm further reduces the computational complexity. Finally, based on the S-RCSC decoding algorithm, we propose a corresponding memory efficient decoder architecture, which has better error performance than existing architectures. Besides, our decoder architecture consumes less energy on updating LLRs.Comment: accepted by the IEEE 2015 workshop on signal processing systems (SiPS

Low-complexity Recurrent Neural Network-based Polar Decoder with Weight Quantization Mechanism

Polar codes have drawn much attention and been adopted in 5G New Radio (NR) due to their capacity-achieving performance. Recently, as the emerging deep learning (DL) technique has breakthrough achievements in many fields, neural network decoder was proposed to obtain faster convergence and better performance than belief propagation (BP) decoding. However, neural networks are memory-intensive and hinder the deployment of DL in communication systems. In this work, a low-complexity recurrent neural network (RNN) polar decoder with codebook-based weight quantization is proposed. Our test results show that we can effectively reduce the memory overhead by 98% and alleviate computational complexity with slight performance loss.Comment: 5 pages, accepted by the 2019 International Conference on Acoustics, Speech, and Signal Processing (ICASSP

Data-Driven Ensembles for Deep and Hard-Decision Hybrid Decoding

Ensemble models are widely used to solve complex tasks by their decomposition into multiple simpler tasks, each one solved locally by a single member of the ensemble. Decoding of error-correction codes is a hard problem due to the curse of dimensionality, leading one to consider ensembles-of-decoders as a possible solution. Nonetheless, one must take complexity into account, especially in decoding. We suggest a low-complexity scheme where a single member participates in the decoding of each word. First, the distribution of feasible words is partitioned into non-overlapping regions. Thereafter, specialized experts are formed by independently training each member on a single region. A classical hard-decision decoder (HDD) is employed to map every word to a single expert in an injective manner. FER gains of up to 0.4dB at the waterfall region, and of 1.25dB at the error floor region are achieved for two BCH(63,36) and (63,45) codes with cycle-reduced parity-check matrices, compared to the previous best result of the paper "Active Deep Decoding of Linear Codes"

Deep Learning-based Polar Code Design

In this work, we introduce a deep learning-based polar code construction algorithm. The core idea is to represent the information/frozen bit indices of a polar code as a binary vector which can be interpreted as trainable weights of a neural network (NN). For this, we demonstrate how this binary vector can be relaxed to a soft-valued vector, facilitating the learning process through gradient descent and enabling an efficient code construction. We further show how different polar code design constraints (e.g., code rate) can be taken into account by means of careful binary-to-soft and soft-to-binary conversions, along with rate-adjustment after each learning iteration. Besides its conceptual simplicity, this approach benefits from having the "decoder-in-the-loop", i.e., the nature of the decoder is inherently taken into consideration while learning (designing) the polar code. We show results for belief propagation (BP) decoding over both AWGN and Rayleigh fading channels with considerable performance gains over state-of-the-art construction schemes.Comment: Allerton201

Deep Unfolding for Communications Systems: A Survey and Some New Directions

Deep unfolding is a method of growing popularity that fuses iterative optimization algorithms with tools from neural networks to efficiently solve a range of tasks in machine learning, signal and image processing, and communication systems. This survey summarizes the principle of deep unfolding and discusses its recent use for communication systems with focus on detection and precoding in multi-antenna (MIMO) wireless systems and belief propagation decoding of error-correcting codes. To showcase the efficacy and generality of deep unfolding, we describe a range of other tasks relevant to communication systems that can be solved using this emerging paradigm. We conclude the survey by outlining a list of open research problems and future research directions.Comment: IEEE Workshop on Signal Processing Systems (SiPS) 2019, special session on "Practical Machine-Learning-Aided Communications Systems.

Syndrome-Enabled Unsupervised Learning for Neural Network-Based Polar Decoder and Jointly Optimized Blind Equalizer

Recently, the syndrome loss has been proposed to achieve "unsupervised learning" for neural network-based BCH/LDPC decoders. However, the design approach cannot be applied to polar codes directly and has not been evaluated under varying channels. In this work, we propose two modified syndrome losses to facilitate unsupervised learning in the receiver. Then, we first apply it to a neural network-based belief propagation (BP) polar decoder. With the aid of CRC-enabled syndrome loss, the BP decoder can even outperform conventional supervised learning methods in terms of block error rate. Secondly, we propose a jointly optimized syndrome-enabled blind equalizer, which can avoid the transmission of training sequences and achieve global optimum with 1.3 dB gain over non-blind minimum mean square error (MMSE) equalizer.Comment: 12 pages, 13 figures, 3 tables. Published in IEEE Journal on Emerging and Selected Topics in Circuits and System

Representation-Oblivious Error Correction by Natural Redundancy

Storage systems have a strong need for substantially improving their error correction capabilities, especially for long-term storage where the accumulating errors can exceed the decoding threshold of error-correcting codes (ECCs). In this work, a new scheme is presented that uses deep learning to perform soft decoding for noisy files based on their natural redundancy. The soft decoding result is then combined with ECCs for substantially better error correction performance. The scheme is representation-oblivious: it requires no prior knowledge on how data are represented (e.g., mapped from symbols to bits, compressed, and combined with meta data) in different types of files, which makes the solution more convenient to use for storage systems. Experimental results confirm that the scheme can substantially improve the ability to recover data for different types of files even when the bit error rates in the files have significantly exceeded the decoding threshold of the ECC.Comment: 7 pages, 5 figures, submitted to IEEE International Conference on Communications-201

Low-Latency SC Decoder Architectures for Polar Codes

Nowadays polar codes are becoming one of the most favorable capacity achieving error correction codes for their low encoding and decoding complexity. However, due to the large code length required by practical applications, the few existing successive cancellation (SC) decoder implementations still suffer from not only the high hardware cost but also the long decoding latency. This paper presents novel several approaches to design low-latency decoders for polar codes based on look-ahead techniques. Look-ahead techniques can be employed to reschedule the decoding process of polar decoder in numerous approaches. However, among those approaches, only well-arranged ones can achieve good performance in terms of both latency and hardware complexity. By revealing the recurrence property of SC decoding chart, the authors succeed in reducing the decoding latency by 50% with look-ahead techniques. With the help of VLSI-DSP design techniques such as pipelining, folding, unfolding, and parallel processing, methodologies for four different polar decoder architectures have been proposed to meet various application demands. Sub-structure sharing scheme has been adopted to design the merged processing element (PE) for further hardware reduction. In addition, systematic methods for construction refined pipelining decoder (2nd design) and the input generating circuits (ICG) block have been given. Detailed gate-level analysis has demonstrated that the proposed designs show latency advantages over conventional ones with similar hardware cost

perm2vec: Graph Permutation Selection for Decoding of Error Correction Codes using Self-Attention

Error correction codes are an integral part of communication applications, boosting the reliability of transmission. The optimal decoding of transmitted codewords is the maximum likelihood rule, which is NP-hard due to the curse of dimensionality. For practical realizations, sub-optimal decoding algorithms are employed; yet limited theoretical insights prevent one from exploiting the full potential of these algorithms. One such insight is the choice of permutation in permutation decoding. We present a data-driven framework for permutation selection, combining domain knowledge with machine learning concepts such as node embedding and self-attention. Significant and consistent improvements in the bit error rate are introduced for all simulated codes, over the baseline decoders. To the best of the authors' knowledge, this work is the first to leverage the benefits of the neural Transformer networks in physical layer communication systems