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In this article, we give the construction of new four-dimensional signal constellations in the Euclidean space, which represent a certain combination of binary frequency-shift keying (BFSK) and M-ary amplitude-phase-shift keying (MAPSK). Description of such signals and the formulas for calculating the minimum squared Euclidean distance are presented. We have developed an analytic building method for even and odd values of M. Hence, no computer search and no heuristic methods are required. The new optimized BFSK-MAPSK (M = 5,6,···,16) signal constructions are built for the values of modulation indexes h =0.1,0.15,···,0.5 and their parameters are given. The results of computer simulations are also provided. Based on the obtained results we can conclude, that BFSK-MAPSK systems outperform similar four-dimensional systems both in terms of minimum squared Euclidean distance and simulated symbol error rate.
Side Channel Attack Resistance of the Elliptic Curve Point Multiplication using Eisenstein Integers
(2020)
Asymmetric cryptography empowers secure key exchange and digital signatures for message authentication. Nevertheless, consumer electronics and embedded systems often rely on symmetric cryptosystems because asymmetric cryptosystems are computationally intensive. Besides, implementations of cryptosystems are prone to side-channel attacks (SCA). Consequently, the secure and efficient implementation of asymmetric cryptography on resource-constrained systems is demanding. In this work, elliptic curve cryptography is considered. A new concept for an SCA resistant calculation of the elliptic curve point multiplication over Eisenstein integers is presented and an efficient arithmetic over Eisenstein integers is proposed. Representing the key by Eisenstein integer expansions is beneficial to reduce the computational complexity and the memory requirements of an SCA protected implementation.
The reliability of flash memories suffers from various error causes. Program/erase cycles, read disturb, and cell to cell interference impact the threshold voltages and cause bit errors during the read process. Hence, error correction is required to ensure reliable data storage. In this work, we investigate the bit-labeling of triple level cell (TLC) memories. This labeling determines the page capacities and the latency of the read process. The page capacity defines the redundancy that is required for error correction coding. Typically, Gray codes are used to encode the cell state such that the codes of adjacent states differ in a single digit. These Gray codes minimize the latency for random access reads but cannot balance the page capacities. Based on measured voltage distributions, we investigate the page capacities and propose a labeling that provides a better rate balancing than Gray labeling.
Soft-input decoding of concatenated codes based on the Plotkin construction and BCH component codes
(2020)
Low latency communication requires soft-input decoding of binary block codes with small to medium block lengths.
In this work, we consider generalized multiple concatenated (GMC) codes based on the Plotkin construction. These codes are similar to Reed-Muller (RM) codes. In contrast to RM codes, BCH codes are employed as component codes. This leads to improved code parameters. Moreover, a decoding algorithm is proposed that exploits the recursive structure of the concatenation. This algorithm enables efficient soft-input decoding of binary block codes with small to medium lengths. The proposed codes and their decoding achieve significant performance gains compared with RM codes and recursive GMC decoding.
This paper proposes a novel transmission scheme for generalized multistream spatial modulation. This new approach uses one Mannheim error correcting codes over Gaussian or Eisenstein integers as multidimensional signal constellations. These codes enable a suboptimal decoding strategy with near maximum likelihood performance for transmission over the additive white Gaussian noise channel. In this contribution, this decoding algorithm is generalized to the detection for generalized multistream spatial modulation. The proposed method can outperform conventional generalized multistream spatial modulation with respect to decoding performance, detection complexity, and spectral efficiency.
Reed-Muller (RM) codes have recently regained some interest in the context of low latency communications and due to their relation to polar codes. RM codes can be constructed based on the Plotkin construction. In this work, we consider concatenated codes based on the Plotkin construction, where extended Bose-Chaudhuri-Hocquenghem (BCH) codes are used as component codes. This leads to improved code parameters compared to RM codes. Moreover, this construction is more flexible concerning the attainable code rates. Additionally, new soft-input decoding algorithms are proposed that exploit the recursive structure of the concatenation and the cyclic structure of the component codes. First, we consider the decoding of the cyclic component codes and propose a low complexity hybrid ordered statistics decoding algorithm. Next, this algorithm is applied to list decoding of the Plotkin construction. The proposed list decoding approach achieves near-maximum-likelihood performance for codes with medium lengths. The performance is comparable to state-of-the-art decoders, whereas the complexity is reduced.
In this letter, we present an approach to building a new generalized multistream spatial modulation system (GMSM), where the information is conveyed by the two active antennas with signal indices and using all possible active antenna combinations. The signal constellations associated with these antennas may have different sizes. In addition, four-dimensional hybrid frequency-phase modulated signals are utilized in GMSM. Examples of GMSM systems are given and computer simulation results are presented for transmission over Rayleigh and deep Nakagami- m flat-fading channels when maximum-likelihood detection is used. The presented results indicate a significant improvement of characteristics compared to the best-known similar systems.
The performance and reliability of non-volatile NAND flash memories deteriorate as the number of program/erase cycles grows. The reliability also suffers from cell to cell interference, long data retention time, and read disturb. These processes effect the read threshold voltages. The aging of the cells causes voltage shifts which lead to high bit error rates (BER) with fixed pre-defined read thresholds. This work proposes two methods that aim on minimizing the BER by adjusting the read thresholds. Both methods utilize the number of errors detected in the codeword of an error correction code. It is demonstrated that the observed number of errors is a good measure for the voltage shifts and is utilized for the initial calibration of the read thresholds. The second approach is a gradual channel estimation method that utilizes the asymmetrical error probabilities for the one-to-zero and zero-to-one errors that are caused by threshold calibration errors. Both methods are investigated utilizing the mutual information between the optimal read voltage and the measured error values.
Numerical results obtained from flash measurements show that these methods reduce the BER of NAND flash memories significantly.
Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multilevel Coding
(2021)
The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.
Error correction coding for optical communication and storage requires high rate codes that enable high data throughput and low residual errors. Recently, different concatenated coding schemes were proposed that are based on binary BCH codes with low error correcting capabilities. In this work, low-complexity hard- and soft-input decoding methods for such codes are investigated. We propose three concepts to reduce the complexity of the decoder. For the algebraic decoding we demonstrate that Peterson's algorithm can be more efficient than the Berlekamp-Massey algorithm for single, double, and triple error correcting BCH codes. We propose an inversion-less version of Peterson's algorithm and a corresponding decoding architecture. Furthermore, we propose a decoding approach that combines algebraic hard-input decoding with soft-input bit-flipping decoding. An acceptance criterion is utilized to determine the reliability of the estimated codewords. For many received codewords the stopping criterion indicates that the hard-decoding result is sufficiently reliable, and the costly soft-input decoding can be omitted. To reduce the memory size for the soft-values, we propose a bit-flipping decoder that stores only the positions and soft-values of a small number of code symbols. This method significantly reduces the memory requirements and has little adverse effect on the decoding performance.
Code-based cryptography is a promising candidate for post-quantum public-key encryption. The classic McEliece system uses binary Goppa codes, which are known for their good error correction capability. However, the key generation and decoding procedures of the classic McEliece system have a high computation complexity. Recently, q-ary concatenated codes over Gaussian integers were proposed for the McEliece cryptosystem together with the one-Mannheim error channel, where the error values are limited to Mannheim weight one. For this channel, concatenated codes over Gaussian integers achieve a higher error correction capability than maximum distance separable (MDS) codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding. This work proposes an improved construction for codes over Gaussian integers. These generalized concatenated codes extent the rate region where the work factor is beneficial compared to MDS codes. They allow for shorter public keys for the same level of security as the classic Goppa codes. Such codes are beneficial for lightweight code-based cryptosystems.
Large-scale quantum computers threaten today's public-key cryptosystems. The code-based McEliece and Niederreiter cryptosystems are among the most promising candidates for post-quantum cryptography. Recently, a new class of q-ary product codes over Gaussian integers together with an efficient decoding algorithm were proposed for the McEliece cryptosystems. It was shown that these codes achieve a higher work factor for information-set decoding attacks than maximum distance separable (MDS) codes with comparable length and dimension. In this work, we adapt this q-ary product code construction to codes over Eisenstein integers. We propose a new syndrome decoding method which is applicable for Niederreiter cryptosystems. The code parameters and work factors for information-set decoding are comparable to codes over Gaussian integers. Hence, the new construction is not favorable for the McEliece system. Nevertheless, it is beneficial for the Niederreiter system, where it achieves larger message lengths. While the Niederreiter and McEliece systems have the same level of security, the Niederreiter system can be advantageous for some applications, e.g., it enables digital signatures. The proposed coding scheme is interesting for lightweight Niederreiter cryptosystems and embedded security due to the short code lengths and low decoding complexity.
Acoustic Echo Cancellation (AEC) plays a crucial role in speech communication devices to enable full-duplex communication. AEC algorithms have been studied extensively in the literature. However, device specific details like microphone or loudspeaker configurations are often neglected, despite their impact on the echo attenuation or near-end speech quality. In this work, we propose a method to investigate different loudspeaker-microphone configurations with respect to their contribution to the overall AEC performance. A generic AEC system consisting of an adaptive filter and a Wiener post filter is used for a fair comparison between different setups. We propose the near-end-to-residual-echo ratio (NRER) and the attenuation-of-near-end (AON) as quality measures for the full-duplex AEC performance.
The encoding of antenna patterns with generalized spatial modulation as well as other index modulation techniques require w-out-of-n encoding where all binary vectors of length n have the same weight w. This constant-weight property cannot be obtained by conventional linear coding schemes. In this work, we propose a new class of constant-weight codes that result from the concatenation of convolutional codes with constant-weight block codes. These constant-weight convolutional codes are nonlinear binary trellis codes that can be decoded with the Viterbi algorithm. Some constructed constant-weight convolutional codes are optimum free distance codes. Simulation results demonstrate that the decoding performance with Viterbi decoding is close to the performance of the best-known linear codes. Similarly, simulation results for spatial modulation with a simple on-off keying show a significant coding gain with the proposed coded index modulation scheme.
List decoding for concatenated codes based on the Plotkin construction with BCH component codes
(2021)
Reed-Muller codes are a popular code family based on the Plotkin construction. Recently, these codes have regained some interest due to their close relation to polar codes and their low-complexity decoding. We consider a similar code family, i.e., the Plotkin concatenation with binary BCH component codes. This construction is more flexible regarding the attainable code parameters. In this work, we consider a list-based decoding algorithm for the Plotkin concatenation with BCH component codes. The proposed list decoding leads to a significant coding gain with only a small increase in computational complexity. Simulation results demonstrate that the Plotkin concatenation with the proposed decoding achieves near maximum likelihood decoding performance. This coding scheme can outperform polar codes for moderate code lengths.
Method and device for error correction coding based on high-rate generalized concatenated codes
(2017)
Field error correction coding is particularly suitable for applications in non-volatile flash memories. We describe a method for error correction encoding of data to be stored in a memory device, a corresponding method for decoding a codeword matrix resulting from the encoding method, a coding device, and a computer program for performing the methods on the coding device, using a new construction for high-rate generalized concatenated (GC) codes. The codes, which are well suited for error correction in flash memories for high reliability data storage, are constructed from inner nested binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer codes, preferably Reed-Solomon (RS) codes. For the inner codes extended BCH codes are used, where only single parity-check codes are applied in the first level of the GC code. This enables high-rate codes.
A soft input decoding method and a decoder for generalized concatenated (GC) codes. The GC codes are constructed from inner nested block codes, such as binary Bose-Chaudhuri-Hocquenghem, BCH, codes and outer codes, such as Reed-Solomon, RS, codes. In order to enable soft input decoding for the inner block codes, a sequential stack decoding algorithm is used. Ordinary stack decoding of binary block codes requires the complete trellis of the code. In one aspect, the present invention applies instead a representation of the block codes based on the trellises of supercodes in order to reduce the memory requirements for the representation of the inner codes. This enables an efficient hardware implementation. In another aspect, there is provided a soft input decoding method and device employing a sequential stack decoding algorithm in combination with list-of-two decoding which is particularly well suited for applications that require very low residual error rates.
Embodiments are generally related to the field of channel and source coding of data to be sent over a channel, such as a communication link or a data memory. Some specific embodiments are related to a method of encoding data for transmission over a channel, a corresponding decoding method, a coding device for performing one or both of these methods and a computer program comprising instructions to cause said coding device to perform one or both of said methods.
Automotive computing applications like AI databases, ADAS, and advanced infotainment systems have a huge need for persistent memory. This trend requires NAND flash memories designed for extreme automotive environments. However, the error probability of NAND flash memories has increased in recent years due to higher memory density and production tolerances. Hence, strong error correction coding is needed to meet automotive storage requirements. Many errors can be corrected by soft decoding algorithms. However, soft decoding is very resource-intensive and should be avoided when possible. NAND flash memories are organized in pages, and the error correction codes are usually encoded page-wise to reduce the latency of random reads. This page-wise encoding does not reach the maximum achievable capacity. Reading soft information increases the channel capacity but at the cost of higher latency and power consumption. In this work, we consider cell-wise encoding, which also increases the capacity compared to page-wise encoding. We analyze the cell-wise processing of data in triple-level cell (TLC) NAND flash and show the performance gain when using Low-Density Parity-Check (LDPC) codes. In addition, we investigate a coding approach with page-wise encoding and cell-wise reading.
Large persistent memory is crucial for many applications in embedded systems and automotive computing like AI databases, ADAS, and cutting-edge infotainment systems. Such applications require reliable NAND flash memories made for harsh automotive conditions. However, due to high memory densities and production tolerances, the error probability of NAND flash memories has risen. As the number of program/erase cycles and the data retention times increase, non-volatile NAND flash memories' performance and dependability suffer. The read reference voltages of the flash cells vary due to these aging processes. In this work, we consider the issue of reference voltage adaption. The considered estimation procedure uses shallow neural networks to estimate the read reference voltages for different life-cycle conditions with the help of histogram measurements. We demonstrate that the training data for the neural networks can be enhanced by using shifted histograms, i.e., a training of the neural networks is possible based on a few measurements of some extreme points used as training data. The trained neural networks generalize well for other life-cycle conditions.