Shorter Codeword Length Research Articles

Adaptive Segmented Aggregation and Rate Assignment Techniques for Flexible-Length Polar Codes.

Polar codes have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provable capacity achieving capability. They have been standardized to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The conventional approach to generate polar codes is to recursively use 2×2 kernels and polarize channel capacities. This approach however, has a limitation of only having the ability to generate codewords of length Norig=2n form. In order to mitigate this limitation, multiple techniques have been developed, e.g., polarization kernels of larger sizes, multi-kernel polar codes, and downsizing techniques like puncturing or shortening. However, the availability of so many design options and parameters, in turn makes the choice of design parameters quite challenging. In this paper, the authors propose a novel polar code construction technique called Adaptive Segmented Aggregation which generates polar codewords of any arbitrary codeword length. This approach involves dividing the entire codeword into smaller segments that can be independently encoded and decoded, thereby aggregated for channel processing. Additionally a rate assignment methodology has been derived for the proposed technique, that is tuned to the design requirement.

An optimal channel coding scheme for high-speed data communication

This paper presents a novel channel-coding scheme called Binary to Gray Code Conversion based Error Detection and Correction (BGCC-EDAC) code to correct the maximum number of errors that occur at the receiving end during wireless transmission. The proposed method adds the parity bit using Binary to Gray Code Conversion (BGCC). The decoding algorithm proposed in this paper will not perform the regular Gray-to-Binary code conversion. Instead, a novel decoding algorithm is proposed to detect and correct most of the errors in the message bits. It will require retransmission only if any parity error occurs. The efficiency of the proposed channel code stems from its redundant bit calculation and decoding algorithm. The simulation results using Cadence 90 nm technology show that the proposed BGCC-EDAC code consumes low power, less area, and less propagation delay. Simulation analysis using MATLAB reveals that the proposed BGCC-EDAC outperforms the existing error-correcting codes in terms of BER performance. This makes the proposed BGCC-EDAC code, with a shorter code word length, highly suitable for high-speed, reliable data communications.

On the Effects of Kernel Configuration in Multi-Kernel Polar Codes

Polar codes are a relatively new family of linear block codes which have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provably capacity achieving capability. They have been proposed to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The basic approach introduced by Arikan can only be used to generate polar codes of length , . To overcome this limitation, polarization kernels of size larger than (like , , and so on), have already been proposed in the literature. Additionally, kernels of different sizes can also be combined together to generate multi-kernel polar codes, further improving the flexibility of codeword lengths. These techniques undoubtedly improve the usability of polar codes for various practical implementations. However, with the availability of so many design options and parameters, designing polar codes that are optimally tuned to specific underlying system requirements becomes extremely challenging, since a variation in system parameters can result in a different choice of polarization kernel. This necessitates a structured design technique for optimal polarization circuits. We developed the DTS-parameter to quantify the best rate-matched polar codes. Thereafter, we developed and formalized a recursive technique to design polarization kernels of higher order from component smaller order. A scaled version of the DTS-parameter, namely SDTS-parameter (denoted by the symbol in this article) was used for the analytical assessment of this construction technique and validated for single-kernel polar codes. In this paper, we aim to extend the analysis of the aforementioned SDTS parameter for multi-kernel polar codes and validate their applicability in this domain as well.

Source Symbol Purging-Based Distributed Conditional Arithmetic Coding.

A distributed arithmetic coding algorithm based on source symbol purging and using the context model is proposed to solve the asymmetric Slepian–Wolf problem. The proposed scheme is to make better use of both the correlation between adjacent symbols in the source sequence and the correlation between the corresponding symbols of the source and the side information sequences to improve the coding performance of the source. Since the encoder purges a part of symbols from the source sequence, a shorter codeword length can be obtained. Those purged symbols are still used as the context of the subsequent symbols to be encoded. An improved calculation method for the posterior probability is also proposed based on the purging feature, such that the decoder can utilize the correlation within the source sequence to improve the decoding performance. In addition, this scheme achieves better error performance at the decoder by adding a forbidden symbol in the encoding process. The simulation results show that the encoding complexity and the minimum code rate required for lossless decoding are lower than that of the traditional distributed arithmetic coding. When the internal correlation strength of the source is strong, compared with other DSC schemes, the proposed scheme exhibits a better decoding performance under the same code rate.

Miscorrection Mitigation for Generalized Integrated Interleaved BCH Codes

The generalized integrated interleaved (GII) codes nest BCH sub-codewords to form codewords of more powerful BCH codes. They can achieve hyper-throughput decoding with excellent error-correcting performance and are among the most suitable codes for next-generation memories. However, the new storage class memories require high code rate and short codeword length. In this case, the sub-codewords have small correction capability. The miscorrections of the sub-codewords lead to much more severe degradation on the GII error-correcting performance compared to the miscorrections of classic BCH codes. This letter investigates the miscorrections in GII decoding. Three low-complexity methods are developed to identify and mitigate the miscorrections by utilizing the nested syndromes, adding one single parity to each sub-codeword, and keeping track of the error locator polynomial degree. Besides, formulas for estimating the miscorrection rates are given. Using the proposed mitigation methods, the actual GII decoding performance becomes very close to that of the case without any miscorrections.

Deep Learning-Based Decoding of Linear Block Codes for Spin-Torque Transfer Magnetic Random Access Memory

Thanks to its superior features of fast read/write speed and low power consumption, spin-torque transfer magnetic random access memory (STT-MRAM) has become a promising non-volatile memory (NVM) technology that is suitable for many applications. However, the reliability of STT-MRAM is seriously affected by the variation of the memory fabrication process and the working temperature, and the later will lead to an unknown offset of the channel. Hence, there is a pressing need to develop more effective error correction coding techniques to tackle these imperfections and improve the reliability of STT-MRAM. In this work, we propose, for the first time, the application of deep-learning (DL)-based algorithms and techniques to improve the decoding performance of linear block codes with short codeword lengths for STT-MRAM. We formulate the belief propagation (BP) decoding of linear block code as a neural network (NN) and propose a novel neural normalized-offset reliability-based min-sum (RB-MS) (NNORB-MS) decoding algorithm. We successfully apply our proposed decoding algorithm to the STT-MRAM channel through channel symmetrization to overcome the channel asymmetry. We also propose an NN-based soft information generation method (SIGM) to take into account the unknown offset of the channel. Simulation results demonstrate that our proposed NNORB-MS decoding algorithm can achieve significant performance gain over both the hard-decision decoding (HDD) and the regular RB-MS decoding algorithm, for cases without and with the unknown channel offset. Moreover, the decoder structure and time complexity of the NNORB-MS algorithm remain similar to those of the regular RB-MS algorithm.

New Locally Correctable Codes Based on Projective Reed–Muller Codes

Locally decodable codes and locally correctable codes (LCCs) have several important applications, such as private information retrieval, secure multiparty computation, and circuit lower bounds. Three major parameters are considered in LCCs: query complexity, message length, and codeword length. The most familiar LCCs in the regime of low query complexity are the generalized Reed–Muller (GRM) codes. However, it has not previously been determined whether there exist codes that have shorter codeword lengths than GRM codes with the same query complexity and message length. In this paper, we show that the projective Reed–Muller (PRM) codes are such LCCs for some parameters. The GRM code is specified by the alphabet size $q$ , the number of variables $m$ , and the degree $d$ , where $d\le q-2$ . When $d= q-2$ and $q-1$ is a power of a prime, we prove that there exists a PRM code with shorter codeword length than the GRM code with the same query complexity and message length. We also present for these PRM codes a perfectly smooth local decoder to recover a symbol in a codeword by accessing not more than $q$ symbols at the coordinates of the codeword.

Evaluation of a Joint Detector Demodulator Decoder (JDDD) Performance With Modulation Schemes Specified in the DVB-S2 Standard

With the rapid increase in the growth of digital broadcast technologies, there has been a demand for high data rate, power, and bandwidth efficient transmission, which, today, is aided by the high-performing low-density parity check (LDPC) codes. In this paper, we explore a novel alternative scheme to the LDPC decoder that performs detection, demodulation, and decoding jointly, coined the joint detector demodulator decoder (JDDD). We test the JDDD through simulation using the modulation schemes defined in the DVB-S2 satellite broadcasting standard. The JDDD is a more recently developed algorithm to the sum–product algorithm (SPA) that is used in decoding the LDPC codes. The JDDD is optimal over a modulated additive white Gaussian noise/intersymbol interference (ISI) channel when resources are sufficient, with the main constraint limiting the algorithm being the availability of computing resources. In this paper, we compare the performance of the system using the JDDD against that of the LDPC decoder. The main result is that the JDDD is able to outperform the iterative detector decoder (IDD) at shorter codeword lengths, when resource requirement is smaller, while the increase in computational requirements tends to favor the IDD at longer CWLs.

Low-complexity LDPC-convolutional codes based on cyclically shifted identity matrices

In this study, a construction methodology for low-density parity-check convolutional codes (LDPC-CCs) ensembles based on cyclically shifted identity matrices is proposed. The proposed method directly generates the syndrome former matrices according to the specified code parameters and constraints i.e., code-rate, degree-distribution, constraint length, period and memory, in contrast to the majority of the available approaches that produce relevant error-correcting codes based on either block ones, protographs or spatially-coupled type of codes. Simulation results show that the constructed ensembles demonstrate advanced error-correcting capability of up to 0.2 dB in terms of frame-error and bit-error rates at the convergence region, when compared with the performance of error-correcting schemes adopted by various communication standards, with equivalent hardware complexity even at short codeword-lengths. Specifically, the constructed LDPC-CCs have been assessed against the corresponding error-correcting codes used in WiMAX and G.hn standards for wireless and wireline telecommunications, respectively.

Low-complexity LDPC-convolutional codes based on cyclically shifted identity matrices

In this study, a construction methodology for low-density parity-check convolutional codes (LDPC-CCs) ensembles based on cyclically shifted identity matrices is proposed. The proposed method directly generates the syndrome former matrices according to the specified code parameters and constraints i.e., code-rate, degree-distribution, constraint length, period and memory, in contrast to the majority of the available approaches that produce relevant error-correcting codes based on either block ones, protographs or spatially-coupled type of codes. Simulation results show that the constructed ensembles demonstrate advanced error-correcting capability of up to 0.2 dB in terms of frame-error and bit-error rates at the convergence region, when compared with the performance of error-correcting schemes adopted by various communication standards, with equivalent hardware complexity even at short codeword-lengths. Specifically, the constructed LDPC-CCs have been assessed against the corresponding error-correcting codes used in WiMAX and G.hn standards for wireless and wireline telecommunications, respectively.

Low Complexity Parity Check Code for Futuristic Wireless Networks Applications

The Internet of Things refers to the aptitude of remotely connecting and monitoring anything, anytime, and anywhere via futuristic wireless networks. Due to the unreliable wireless links, broadcast nature of wireless transmissions, interference and noisy transmission channels, frequent topology changes, and the various quality of wireless channel, there are challenges in providing high data rate service, high throughput, high packet delivery ratio, low end-to-end delay, and reliable services. In wireless network and real time application systems, low complexity and shorter codeword length in channel coding scheme are preferred. Consequently, in order to address these challenges, we propose a novel error detection and correction codes called the low-complexity parity-check (LCPC) codes with short codeword lengths for futuristic wireless networks applications. The proposed codes have less complexity and lower memory requirement in comparison to turbo and low-density parity-check (LDPC) codes. Simulation results demonstrated that the proposed LCPC codes outperform the Hamming and Reed–Solomon codes, in addition to the renowned LDPC codes. It offers up to 3-dB coding gain.

Efficient approximate message authentication scheme

An approximate message authentication scheme is a primitive that allows a sender Alice to send a source state to a receiver Bob such that the latter is assured of its authenticity, where the source state is considered as authentic if it only undergoes a minor change. Here, the authors propose an efficient scheme for this problem and prove its security under a rigorous model. Our scheme only needs a lightweight computation cost and hence is very efficient. As the authentication message is transmitted over a noisy channel, we also value the channel efficiency (i.e. the coding rate). For a fixed coding method, this is determined by the admissible decoding bit error probability p e . A larger p e admits a shorter codeword length and hence a larger coding rate. It turns out that the p e can be set to be a significantly large constant (determined by the legal distortion level for the source state). Compared with existing schemes, the advantage in p e is evident.

Enhanced Successive Cancellation List Decoding of Polar Codes

Polar codes can be efficiently decoded by the successive cancellation list (SCL) algorithm in which the sorter plays an important role. The sorter, however, is time-consuming. We significantly improve this decoding scheme by developing a new non-sorting direct selection method for the SCL algorithm. In the non-sorting strategy, we design a new selector for choosing the $L$ best surviving paths, which is achieved by exploiting the bitwise comparison of the candidate path metrics. The analysis and simulation results show that, for $M$ -bit length path metrics, our direct selection method with time complexity $O(ML)$ ) is superior to the latest simplified bubble sorter and the quick sorting strategy whose time complexity is $O(ML^{2})$ and $O(ML \log L)$ , respectively. In the scenario of short codeword length and large list size, our selector can remarkably reduce the latency and the hardware resource consumption while not assuming that the metrics in the decoding are sorted.

A Hybrid Decoding Scheme for Short Non-Binary LDPC Codes

In this paper, an iterative soft-decision hybrid decoding algorithm for non-binary low-density parity-check (LDPC) codes with short codeword lengths is proposed. The rationale of the approach is to combine the classical belief propagation (BP) iterative LDPC decoding algorithm with the most reliable basis (MRB) decoding algorithm. This allows to achieve significant performance improvements, with a complexity that, for medium/low error rates, is only slightly higher than that of the BP algorithm alone. The performance improvement with respect to pure BP decoding is up to 0.7 dB at codeword error rate (CER) ≈ 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> . Notably, for a fixed MRB order, hybrid decoding achieves a gain up to 0.5 dB at CER ≈ 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> with respect to BP decoding and MRB decoding used alone.

Distributed Joint Source-Channel Coding Using Unequal Error Protection LDPC Codes

This paper presents a general approach to designing distributed joint source channel (DJSC) codes with arbitrary rates for communication of a pair of correlated binary sources over noisy channels. In this approach, both distributed compression and channel error correction are simultaneously achieved by transmitting, for each source, a fraction of the information bits together with the parity bits of a systematic channel code. This approach is shown to be asymptotically optimal, i.e., any rate-pair in the achievable rate-region can be approached as the codeword length is increased. The practical realization of such a code requi res the design of a pair of channel codes with unequal error protection (UEP) properties determined by the inter-source correlation and the channel capacity available to each source. Towards this end, a linear programming based procedure for jointly optimizing the degree profiles of a pair of irregular LDPC codes to achieve the required UEP properties is presented. Experimental results obtained with both binary symmetric channels and binary-input Gaussian channels are presented, which demonstrate that the proposed UEP-DJSC codes can significantly outperform separate source-channel codes, as well as previously reported joint source-channel coding schemes, particularly for short codeword lengths.

Channel Capacity and Soft-Decision Decoding of LDPC Codes for Spin-Torque Transfer Magnetic Random Access Memory (STT-MRAM)

Spin-torque transfer magnetic random access memory (STT-MRAM) has emerged as a promising non-volatile memory (NVM) technology, featuring compelling advantages in scalability, speed, endurance, and power consumption. In this paper, we focus on large-capacity stand-alone STT-MRAM, and investigate the channel capacity and the viability of applying low-density parity-check (LDPC) codes with soft-decision decoding to correct the memory cell errors and improve the storage density of STT-MRAM. We propose to use LDPC codes with short codeword lengths, with the reliability-based min-sum (RB-MS) algorithm for decoding. Furthermore, we propose to use the capacity-maximization criterion to design the quantizer and minimize the number of quantization bits. Simulation results demonstrate the potential of applying short-block-length LDPC codes with soft-decision decoding to improve the yield and push the scaling limitation of STT-MRAM.

The construction of binary Huffman equivalent codes with a greater number of synchronising codewords

An inherent problem with a Variable–Length Code (VLC) is that even a single bit error can cause a loss of synchronisation, and thus lead to error propagation. Codeword synchronisation has been extensively studied as a means to overcome this drawback and efficiently stop error propagation. In this paper, we first present the sufficient and necessary conditions for the existence of binary Huffman equivalent codes with the shortest, or at most two shortest, synchronising codeword(s) of length m + 1, where m (>1) is the shortest codeword length. Next, based on the results, we propose a unified approach for constructing each of these binary Huffman equivalent codes with the shortest, or at most two shortest, synchronising codeword(s) of length m + 1, if such a code exists for a given length vector.

Multiple‐encoder layered space–time–frequency architecture with QR decomposition and M‐algorithm maximum‐likelihood detection

ABSTRACTIn this paper, to achieve the available spatial, temporal, and frequency diversities, and also to make the system implementation feasible for high‐speed orthogonal frequency division multiplexing multiple‐input multiple‐output multiplexing, a novel layered space–time–frequency (LSTF) transmitter architecture with multiple channel encoders is proposed with each independently coded information sub‐stream being threaded in the three‐dimensional (3‐D) space–time–frequency transmission resource array. The performance of a non‐iterative receiver consisting of a maximum‐likelihood detector with QR decomposition and M‐algorithm maximum‐likelihood detection is exploited, by employing irregular low‐density parity‐check code as the channel code. The threaded distribution of each coded information sub‐stream in the proposed LSTF architecture makes it achieve the spatial, temporal, and frequency diversities the same as the conventional single‐encoder LSTF architecture where coding is applied across the whole information stream, and simulation results show that the performance of the proposed multiple‐encoder LSTF architecture is very close to that of the conventional single‐encoder LSTF architecture. However, because of the use of multiple parallel encoders/decoders with a shorter codeword length, the proposed LSTF architecture has much lower hardware processing speed and complexity than that of the conventional LSTF architecture. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Multiple LDPC-Encoder Layered Space-Time-Frequency Architectures for OFDM MIMO Multiplexing

This paper is focused on the study of layered space-time-frequency (LSTF) architectures with channel coding for orthogonal frequency division multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing systems for high speed wireless communications over a frequency-selective fading channel. In order to achieve the available spatial, temporal and frequency diversities, and also make the system implementation feasible for high speed OFDM MIMO multiplexing, a novel LSTF architecture with multiple channel encoders is proposed with each independently coded layer being threaded in the three-dimensional space-time-frequency transmission resource array. Non-iterative receiver is adopted which consists of list sphere detector and irregular low-density parity-check codes as the constituent codes. Simulation results show that the performance of the proposed multiple-encoder LSTF architecture is very close to that of the conventional single-encoder LSTF where coding is applied across the whole information stream. However, due to the use of multiple parallel encoders/decoders with a shorter codeword length, the proposed LSTF architecture has much lower hardware processing speed and complexity than the conventional LSTF.

Iterative Signal Processing for Coded LSTF Architectures

For high speed orthogonal frequency division multiplexing (OFDM) plus multiple-input multiple-output (MIMO) multiplexing, a new coded layered space-time-frequency (LSTF) architecture with iterative signal processing at the receiver is proposed, where each independent codeword is threaded in the three-dimensional (3D) space-time-frequency transmission resource array. The iterative receiver structure is adopted consisting of a joint minimum-mean-square-error soft-interference-cancellation (MMSE-SIC) detector and the maximum a posteriori (MAP) convolutional decoders. Simulation results show that the proposed LSTF architecture can achieve almost the same performance as the LSTF (i.e., LSTF-a) where coding is applied across the whole information stream. However, due to its structure of multiple parallel lower speed encoders/decoders with shorter codeword length, the proposed LSTF architecture can be more easily implemented than the LSTF-a.