Valid Codeword Research Articles

Optimal coefficient selection in Compute and Forward relaying using improved Schnorr–Euchner search algorithm

Physical layer Network Coding with Compute and Forward relaying (CF) is a promising transmission strategy that effectively exploits the natural linear computation of the multiuser wireless channel. In CF relaying, the relay nodes receive a linear combination of transmitted signals with additive noise and each relay node computes an integer linear combination of transmitted codewords which are then forwarded to the destination. After receiving enough independent equations from multiple relay nodes, the destination could decode individual source nodes’ messages. The source nodes use nested lattice codes which have the property that an integer linear combination of codewords is a valid codeword in the codeset. The simulation set up includes a K×K×1 network model with K source nodes, K relay nodes and 1 destination node. Relay nodes are assumed to have perfect channel state information, which is used in computing the linear equation. The success of CF relaying lies in choosing the optimal coefficient in computing the independent linear equation at each relay node. Optimal coefficient selection aims to maximize the computation rate which can be achieved when the equation coefficients are matched with complex-valued channel gains. This paper proposes an iterative algorithm for finding the optimal coefficient vector in the Gaussian integer domain using an improved Schnorr–Euchner search strategy. The performance of the proposed coefficient search algorithm in terms of computation complexity and the achievable rate is compared with other existing algorithms through MATLAB simulation. Simulation is performed for networks with 2, 4, 8 and 16 numbers of source and relay nodes. The proposed coefficient selection algorithm is validated for varying Signal to Noise Power under Rayleigh fading channel environment.

Symbol-level iterative information set decoding of RS codes

This paper presents the implementation of a low-complex iterative symbol-level decoding scheme for Reed-Solomon codes. Most soft-decision iterative decoders for Reed-Solomon codes work on a bit level due to the efficient passing of soft information due to the sparsity of the binary parity check matrix. These decoders yield a good error correction performance, but this comes at the cost of an increased computational complexity resulting from working at a bit-level. This study aims to lower the computational complexity of iterative decoding of Reed-Solomon codes by introducing a high-performance soft-decision iterative Reed-Solomon decoder that works on a symbol-level, in contrast to the bit-level implementation used in most iterative Reed-Solomon decoders. The proposed algorithm utilises soft information to decode while applying information set decoding techniques to an extended parity check matrix. The use of the extended parity check matrix provides additional parity check equations which assist the algorithm in determining the correct information set of symbols used in the decoding process. For a given (n,k) Reed-Solomon code, the algorithm converges to a valid codeword by correctly decoding a specified information set of k symbols from the received vector. Simulations run show the proposed algorithm performs favourably when compared to other symbol-level iterative decoders while working at a relatively lower complexity.

Near-lossless compression scheme using hamming codes for non-textual important regions in document images

Working at Bell Labs in 1950, irritated with error-prone punched card readers, R W Hamming began working on error-correcting codes, which became the most used error-detecting and correcting approach in the field of channel coding in the future. Using this parity-based coding, two-bit error detection and one-bit error correction was achievable. Channel coding was expanded further to correct burst errors in data. Depending upon the use of the number of data bits ‘d’ and parity bits ‘k’ the code is specified as (n, k) code, here ‘n’ is the total length of the code (d+k). It means that 'k' parity bits are required to protect 'd' data bits, which also means that parity bits are redundant if the code word contains no errors. Due to the framed relationship between data bits and parity bits of the valid codewords, the parity bits can be easily computed, and hence the information represented by 'n' bits can be represented by 'd' bits. By removing these unnecessary bits, it is possible to produce the optimal (i.e., shortest length) representation of the image data. This work proposes a digital image compression technique based on Hamming codes. Lossless and near-lossless compression depending upon need can be achieved using several code specifications as mentioned here. The achieved compression ratio, computational cost, and time complexity of the suggested approach with various specifications are evaluated and compared, along with the quality of decompressed images.

Parity-check and G-matrix based intelligent early stopping criterion for belief propagation decoder for polar codes

The error correction performance of Belief Propagation (BP) decoding for polar codes is satisfactory compared with the Successive Cancellation (SC) decoding. Nevertheless, it has to complete a fixed number of iterations, which results in high computational complexity. This necessitates an intelligent identification of successful BP decoding for early termination of the decoding process to avoid unnecessary iterations and minimize the computational complexity of BP decoding. This paper proposes a hybrid technique that combines the “parity-check” with the “G-matrix” to reduce the computational complexity of BP decoder for polar codes. The proposed hybrid technique takes advantage of the parity-check to intelligently identify the valid codeword at an early stage and terminate the BP decoding process, which minimizes the overhead of the G-matrix and reduces the computational complexity of BP decoding. We explore a detailed mechanism incorporating the parity bits as outer code and prove that the proposed hybrid technique minimizes the computational complexity while preserving the BP error correction performance. Moreover, mathematical formulation for the proposed hybrid technique that minimizes the computation cost of the G-matrix is elaborated. The performance of the proposed hybrid technique is validated by comparing it with the state-of-the-art early stopping criteria for BP decoding. Simulation results show that the proposed hybrid technique reduces the iterations by about 90% of BP decoding in a high Signal-to-Noise Ratio (SNR) (i.e., 3.5 ∼ 4 ​dB), and approaches the error correction performance of G-matrix and conventional BP decoder for polar codes.

LDPC Codes Over GF(q) With Alterable Subset for PAPR Reduction in OFDM Systems

Non-binary low-density parity-check (NB-LDPC) codes can exhibit more considerable coding gain than their binary counterparts in numerous cases. To further exploit the benefit of NB-LDPC codes, in this letter, a novel type of non-binary LDPC codes, alterable subset non-binary LDPC (AS-NB-LDPC) codes, is proposed for reducing the peak-to-average power ratio (PAPR) of orthogonal frequency division multiplexing (OFDM) systems. Firstly, we design the alteration scheme of AS-NB-LDPC codes to make <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary LDPC codes generate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> valid codewords through one alterable subset, and the codeword with the minimum PAPR is selected for transmission. Then inspired by the invertible subset for binary LDPC codes, the necessary and sufficient conditions for alterable subset generation are derived. Furthermore, we propose an improved progressive edge growth (PEG) construction method to obtain AS-NB-LDPC codes with favorable frame error rate (FER) performance. Simulation results verify the effectiveness of the proposed NB-LDPC codes.

A Fixed Latency ORBGRAND Decoder Architecture With LUT-Aided Error-Pattern Scheduling

Guessing Random Additive Noise Decoding (GRAND) is a universal decoding algorithm that has been recently proposed as a practical way to perform maximum likelihood decoding. It generates a sequence of possible error patterns and applies them to the received vector, checking if the result is a valid codeword. Ordered reliability bits GRAND (ORBGRAND) improves on GRAND by considering soft information received from the channel. Both GRAND and ORBGRAND have been implemented in hardware, focusing on average performance, sacrificing worst case throughput and latency. In this work, an improved pattern schedule for ORBGRAND is proposed. It provides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt; 0.5$ </tex-math></inline-formula> dB gain over the standard schedule at a block error rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\le 10^{-5}$ </tex-math></inline-formula> , and outperforms more complex GRAND flavors with a fraction of the complexity. The proposed schedule is used within a novel code-agnositic decoder architecture: the decoder guarantees fixed high throughput and low latency, making it attractive for latency-constrained applications. It outperforms the worst-case performance of decoders by orders of magnitude, and outperforms many best-case figures. Decoding a code of length 128, it achieves a throughput of 79.21 Gb/s with 58.49 ns latency, yielding better energy efficiency and comparable area efficiency with respect to the state of the art.

On Decoding of Reed-Muller Codes Using a Local Graph Search

We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge if and only if the Hamming distance between the corresponding codewords equals the minimum distance of the code. The algorithm uses a greedy local search to find a node optimizing a metric, e.g. the correlation between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as a valid codeword is found, leading to an improvement in the average computational complexity of the algorithm. Simulation results for both binary symmetric channel and additive white Gaussian noise channel show that the presented decoder approaches the performance of maximum likelihood decoding for RM codes of length less than 1024 and for the second-order RM codes of length less than 4096. Moreover, it is demonstrated that the considered decoding approach outperforms state-of-the-art decoding algorithms of RM codes with similar computational complexity for a wide range of block lengths and rates.

A hybrid Sphere Decoding for short polar codes using variable step size

In this paper, a hybrid stack Sphere Decoding (SD) for short polar codes based on radius extension and a novel path metric (PM) is proposed. The proposed SD scheme utilizes variable step size obtained from a novel PM, which is a modified PM from a previous work. The proposed method continuously increases the corresponding variable radius until a valid polar codeword is found. Moreover, by introducing a limited size stack, the proposed scheme can avoid node revisiting and achieve a trade-off between performance and complexity. Simulation results show that for short polar codes, compared with existing SD schemes that achieve maximum likelihood (ML) performance, the proposed scheme achieves more complexity reduction without compromising the ML optimality.

Skip-Sliding Window Codes

Constrained coding is used widely in digital communication and storage systems. In this paper, we study a generalized sliding window constraint called the skip-sliding window. A skip-sliding window (SSW) code is defined in terms of the length $L$ of a sliding window, skip length $J$, and cost constraint $E$ in each sliding window. Each valid codeword of length $L + kJ$ is determined by $k+1$ windows of length $L$ where window $i$ starts at $(iJ + 1)$th symbol for all non-negative integers $i$ such that $i \leq k$; and the cost constraint $E$ in each window must be satisfied. In this work, two methods are given to enumerate the size of SSW codes and further refinements are made to reduce the enumeration complexity. Using the proposed enumeration methods, the noiseless capacity of binary SSW codes is determined and observations such as greater capacity than other classes of codes are made. Moreover, some noisy capacity bounds are given. SSW coding constraints arise in various applications including simultaneous energy and information transfer.

Joint Coding Strategy of the Phase Domain and Intensity Domain for Absolute Phase Retrieval

Codewords used to determine fringe orders are significant for absolute phase retrieval based on the phase-coding method. However, because of inevitable interference from image blurring, image noise, and system nonlinearity, it is challenging to significantly increase the number of codewords for high-accuracy phase retrieval. To address this challenge, this article proposes a joint coding strategy to perform the coding of high-density stripes. More specifically, codewords are simultaneously embedded into the phase domain and the intensity domain of composite fringe patterns, which greatly extends the encoding range of valid codewords. Moreover, to improve the anti-interference capability of the proposed coding strategy, a regional correction technique (RCT) is presented to remove jump phase errors induced by the inevitable interference. The results of experiments indicate that the proposed method is effective and accurate for both complex objects and a standard sphere.

Enhanced Quasi-Maximum Likelihood Decoding Based on 2D Modified Min-Sum Algorithm for 5G LDPC Codes

We propose a two-dimensional modified min-sum algorithm for the LDPC codes in the fifth generation (5G) networks standard to approach the error performance of the sum-product algorithm (SPA). In the proposed decoding algorithm, we adopt a partial self-correction method followed by message amplification to improve the reliability of the variable-to-check (V2C) messages. To further approach the performance of the maximum likelihood decoding for 5G short LDPC codes, we propose an enhanced quasi-maximum likelihood (EQML) decoding method. The proposed decoding method performs multiple rounds of decoding tests once the first decoding attempt fails, where the decoder inputs of the selected unreliable variable nodes are modified in each decoding test. A novel node selection method based on the sign fluctuation of V2C messages is proposed for the EQML decoding method. We also present a partial pruning stopping (PPS) rule to reduce the decoding complexity by deactivating part of the decoding tests once a valid codeword is found. A lower bound on the error performance is also derived by using the semi-analytical method. Simulation results show that the EQML decoding method outperforms the SPA with the same decoding complexity and other QML decoding methods, and it approaches the Polyanskiy-Poor-Verdu bound within 0.4 dB.

Low Complexity Bit Reliability and Predication Based Symbol Value Selection Decoding Algorithms for Non-Binary LDPC Codes

The main challenge for hardware implementation of non-binary LDPC decoding is the high computational complexity and large memory requirement. To address this challenge, five new low complexity LDPC decoding algorithms are proposed in this paper. The proposed algorithms are developed specifically towards the low complexity, yet effective, decoding of the NB LDPC codes. The proposed decoding algorithms update, iteratively, the hard decision received vector to search for the valid codeword in the vector space of Galois field ${(GF)}$ . The selection criterion for least reliable symbol positions is based on the information from the failed checks and the reliability information from the Galois field structure as well as from the received channel soft information. To choose the correct value for the candidate symbol, two methods are used. The first method is based on the prediction of the error symbol from the set of Galois field symbols which maximize an objective function. In the second method, individual bits are flipped based on the reliability information obtained from the channel. Algorithms 1 and 2 flip a single symbol per iteration whilst the other three algorithms 3 , 4 and 5 flip multiple symbols in each iteration. The proposed voting based Algorithms 1 , 2 and 5 first short list the unreliable positions using a majority voting scheme and then choose the candidate symbol value from the set of the symbols in ${GF(q)}$ while not violating the field order ${q}$ . These methods simplify the decoding complexity in terms of computation and memory. Results and analysis of these algorithms show an appealing tradeoff between computational complexity and bit error rate performance for NB LDPC codes.

Wireless Multiple Access Network With a Sectional Configured Data Collection System and Codeword Filtering Mechanism

Swarm robotics is an approach to collective robotics with which it is easily possible to complete the tasks that are difficult to do with single robot. The wireless multiple access network contributes significantly in wireless mobile swarm robotics for communication and networking, especially in disaster scenarios. Since the unpredictable impacts may fail the sensing robots or nodes, causing the loss of valuable data, a coding scheme called the Growth codes is designed to increase data persistence. In the wireless multiple access network, a key requirement of data collection in disaster scenarios is to maintain a high data intermediate recovery rate. In the later period of the Growth codes, the large amount of redundant data in the network affects the efficiency of data collection. In this paper, we design and analyze techniques to reduce the number of redundant data, and propose a sectional configured data collection strategy based on the Growth codes, called the SCGC protocol. Setting cache nodes around the sink to filter the redundant codewords, the sink can collect data faster, ensuring a high data intermediate recovery rate. We also design an information update strategy and some constraints to control node overhead. Through the simulations, we show that the proportion of redundant codewords can be reduced by 10-15%, while the negative impact of these valid codewords is not considered in the Growth codes. We also show that the SCGC protocol can improve the data recovery efficiency without affecting the stability of the network or shortening the network lifetime.

A CRC-aided Perturbed Decoding of Polar Codes

We propose a CRC-aided perturbed decoding method for polar codes. Our technique is an iterative process that makes available multiple codeword candidates from a received signal. A CRC decoder is used to assess each decoded sequence. When a decoded sequence fails the CRC a random perturbation noise is added to the received signal, and then the perturbed signal is decoded again by a conventional decoder. This process continues until a valid codeword is found or the threshold for maximum iterations is attained. The simulation results indicate that the proposed scheme has a 0.4dB performance gain over existing schemes at the expense of a minimal increase in complexity. The proposed algorithm has the added advantage of being applicable to most concatenating decoding techniques.

On Tanner Graph Error-Detection of Some Codes due to (123)/ (132)-Avoiding Patterns of AUNU Numbers

We consider some of the codes generated due to the AUNU patterns as studied earlier by the authors (the [5 3 2] linear code, [7 4 2]-linear code and the [14 8 3] codes) . The Parity check matrices of these codes are obtained and used in constructing their Tanner graphs. A Tanner graph has n variable nodes, corresponding to the code bits of the codeword, and m check nodes that represents the parity check equations. An assignment of the code bits of a received vector is now known to be a valid code word if all the parity check equations from the Tanner graph are satisfied otherwise, an error is detected. Moreover, other code properties such as the regularity or otherwise of these codes are also determined from such Tanner graphs.

Low-Latency CRC Calculation in Turbo-Code Decoding

Design of Turbo-Code decoders for the 3GPP LTE/LTE-A standard focuses on achieving a high-throughput of up to 1 Gbps with low decoding latencies at very high code rates. At code rates close to one, as specified in LTE/LTE-A, the decoding process can oscillate. Although the decoder converges to a valid code word after half-iteration i, after half-iteration \(i+1\) the decoded code word is invalid again. To circumvent this, the CRC that is attached to each code word must be evaluated after each half-iteration. We present a generalized architecture for calculating the CRC On-the-fly during both non-interleaved and interleaved half-iterations and thus explicitly taking into account the requirements for low-latency Turbo-Code decoding. Further, we investigate the latency-, and energy savings provided by employing this calculation scheme. Lastly, we give post layout synthesis results for a case study implementation in state-of-the-art 28 nm FDSOI technology.

The 64-Triplet Genetic Code Structure Revisited and Refuted from Combinatorial

Statement of the problem: When molecular biologists observed in 1953 that the sequence of the DNA four bases in the nucleus of a cell influenced the sequence of the twenty amino acids of protein in the cytoplasm, they desired to find a code to account for the correlation, and eventually had the 64 triplet genetic code in 1954 from a mathematician, which is currently in use but not flawless. Methodology and theoretical orientation: The said observation is seen as a natural example of an input/output system, in which the input is the DNA four bases and the output is the sequence of 24 permutations of the four DNA bases constituting the genetic code in the cytoplasm. A combinatorial input/output multiplicative replication system armed with basic permutations computation schemes is now available to produce permutations systematically, such as Square Kinematics Scheme and Successive Collateral Posting Scheme used. Findings: A 24 quadruplet genetic code was produced by each of the two methods with an input set of the DNA four bases. It is shown in the successive collateral posting method that the 64 triplets comprise 40 non-permutations and 24 permutations. The 40 non-permutation triplets are crossed out leaving 24 permutation triplets which are undersized and therefore unqualified to represent the genetic code output sequence from an input set of 4. Conclusion and significance: The 24 quadruplet genetic code is a breakthrough in the Molecular Biologists’ search for a code following their observation which ended up with the 64-triplet genetic code that has no combinatorially valid code word being triplets, instead of quadruplets. Recommendations: These are made towards effective publicity of the new 24 quadruplet genetic code to attract experimental experts to spell it to win adoption in coding application in protein studies for a desired relief to stakeholders in genetics.

Improvement Of Block Product Turbo Coding By Using A New Concept Of Soft Hamming Decoder

The block product turbo code (BPTC) is classified as one of block turbo code concatenation forms. The Hamming code can detect two-bit error and correct one-bit error. The BPTC uses two Hamming codes for "column" coding and "row" coding, it has improved the Hamming code correcting only one error. In addition, the BPTC carries out block interleaving coding for disorganizing the transmission sequence before transmission, so as to avoid burst errors when the signal meets multi-path channel in the channel. This paper will discuss the decoding mechanism of the BPTC and analyze the efficiency of using a soft decoding algorithm in the decoding process. The soft Hamming Decoder is based on error patterns which belong to the same syndrome. It is shown that it is sufficient to investigate error patterns with one and two errors to gain up to 1.2 dB compared to hard decision decoding. Here, we will consider also the error patterns with three errors which belong to the determined syndrome, which increases the gain and improves the quality of the soft-output due to the increased number of comparisons with valid code words, in despite that, it will increase the complexity of the decoding process. The system is based on two Hamming block channel code combinations, which can be similar or different, a block interleaving to construct a BPSK modulation and BPTC coding system in the concept of feedback encoding in turbo code over an AWGN channel. To observe its coding improvement, we present the simulation results for the soft decoding of the BPTC codes of a code word length from 49 bits (using two (7,4) codes) up to 1440 bits (using two (127,120) codes).

Maximally Distant Codes Allocation Using Chemical Reaction Optimization and Ant Colony Optimization Algorithms

Error correcting codes, also known as error controlling codes, are set of codes with redundancy that allows detecting errors. This is quite useful in transmitting data over a noisy channel or when retrieving data from a storage with possible physical defects. The idea is to use a set of code words that are maximally distant from each other, hence reducing the chance of changing a valid codeword to another valid codeword by flipping bits. The problem can be viewed as picking m codes out of 2 n available n- bit combinations, such that the aggregate hamming distance among those codewords is maximized. Due to the large solution spaces of such problems, greedy algorithms are sometimes used to generate quick and dirty solutions. However, modern evolutionary search algorithms like genetic algorithms, swarm particles, gravitational search and others, offer good alternatives, yielding near optimal solutions in exchange for some time. Chemical Reaction Optimization (CRO) has emerged as a new evolutionary algorithm to solve complex optimization problems. This algorithm mimics the molecular interactions towards finding a minimal energy state, which corresponds to an optimal solution for the problem in hand. In this research, we proposed a solution for the maximally distant codes allocation problem, through a binary knapsack mapping and compared the performance with the well established Ant Colony Optimization (ACO) algorithm, which is inspired by the ant's capability to find the shortest path between the nest and source of food. The binary knapsack mapping was used in the two algorithms. Test results showed that the CRO outperformed the ACO in every metric given any time budget.

Invertible Subset QC-LDPC Codes for PAPR Reduction of OFDM Signals

A type of invertible subset low density parity-check (IS-LDPC) codes was proposed to reduce the peak-to-average power ratio for orthogonal frequency-division multiplexing systems with low complexity recently, where the key idea is that each invertible subset can be independently inverted to generate other valid codewords of the LDPC code. In this paper, we firstly extend the idea of invertible subsets to quasi-cyclic LDPC (QC-LDPC) codes, and this new family of QC-LDPC codes is called as invertible subset QC-LDPC (IS-QC-LDPC) codes. Moreover, we propose two types of subset partitioning and deeply discuss the properties of IS-QC-LDPC codes with different subset partitioning. After declaring that interleaved partitioning is more appropriate for IS-QC-LDPC codes than the block partitioning, we propose a novel progressive edge graph-like algorithm to construct good IS-QC-LDPC codes with interleaved partitioning. The simulation results show that the IS-QC-LDPC codes exhibit good error-correcting performance, which is very close to that of the corresponding QC-LDPC codes of the IEEE 802.16e, for a wide range of subset number.