Hadamard Codes Research Articles

On the Rank of [formula omitted]-linear Hadamard Codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s>2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s=3. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once t≥3 is fixed. In this case, the number of nonequivalent such codes is also established.

Shipborne Underwater Acoustic Communication System and Sea Trials with Submersible Shenhai Yongshi

The Shipborne acoustic communication system of the submersible Shenhai Yongshi works in vertical, horizontal and slant channels according to the relative positions. For ease of use, an array combined by a vertical-cone directional transducer and a horizontal-toroid one is installed on the mothership. Improved techniques are proposed to combat adverse channel conditions, such as frequency selectivity, non-stationary ship noise, and Doppler effects of the platform’s nonlinear movement. For coherent modulation, a turbo-coded single-carrier scheme is used. In the receiver, the sparse decision-directed Normalized Least-Mean-Square soft equalizer automatically adjusts the tap pattern and weights according to the multipath structure, the two receivers’ asymmetry, the signal’s frequency selectivity and the noise’s spectrum fluctuation. The use of turbo code in turbo equalization significantly suppresses the error floor and decreases the equalizer’s iteration times, which is verified by both the extrinsic information transfer charts and bit-error-rate performance. For noncoherent modulation, a concatenated error correction scheme of nonbinary convolutional code and Hadamard code is adopted to utilize full frequency diversity. Robust and low-complexity synchronization techniques in the time and Doppler domains are proposed. Sea trials with the submersible to a maximum depth of over 4500 m show that the shipborne communication system performs robustly during the adverse conditions. From the ten-thousand communication records in the 28 dives in 2017, the failure rate of the coherent frames and that of the noncoherent packets are both below 10%, where both synchronization errors and decoding errors are taken into account.

A New High-Efficiency Multilevel Frequency-Modulation Different Chaos Shift Keying Communication System

Considering the large number of delay lines and orthogonality Walsh code sequences in the traditional multilevel frequency-modulation different chaos shift keying (FM-DCSK), in this paper, we propose a new multilevel FM-DCSK, named multilevel M-ary FM-DCSK (MFM-DCSK). The MFM-DCSK transmits multilevel symbols directly in the communication channel without using orthogonality Walsh code sequence. Based on this, the MFM-DCSK can reduce computation complexity and time delay and improve transmission rate dramatically. In this paper, we also evaluate the bit error rate (BER) performance of the MFM-DCSK in an additive white Gaussian noise channel, a multipath Rayleigh fading channel, and a multiuser channel. Compared with the conventional chaos communication algorithms, the MFM-DCSK can improve the integrated performance of the communication algorithm significantly. Moreover, we also define the utility of the chaos communication algorithm, which takes the transmission rate, the number of delay lines, and the BER into consideration. According to this definition, we deduce the optimal value of M for both MFM-DCSK and traditional M-ary FM-DCSK algorithms.

A note on regular subgroups of the automorphism group of the linear Hadamard code

A note on regular subgroups of the automorphism group of the linear Hadamard code

Chaotic sequence‐based MC‐CDMA for 5G

SummaryA unique chaotic spreading sequence‐based MC‐CDMA (multi‐carrier code division multiple accesses) is studied in this article. By employing this spreading sequence, interference effect that is inherent in a multi‐user environment can be lessened by deciding on the spreading series with suitable cross‐correlation properties. The simulation study exhibits that chaotic sequence‐based MC‐CDMA acts well in a multiuser setting, resulting in better attainable error rates. Moreover, results confirm that chaotic sequence‐based MC‐CDMA beats the Walsh Hadamard code spreading‐based MC CDMA system. The inception of MIMO (Multiple Input Multiple Output) scheme leads to the augment in the number of antennas that are present in the transmitting in addition to receiving side. This induces an enhancement in the throughput and the robustness of the communication system. Further, the simulation outcomes exhibit that the suggested technique is far beneficial in the context of combating MAI (multiple access interference) and aids to achieve performance amelioration concerning the utility of approximately 9%.

Hiding Stealth Optical CDMA Signals in Public BPSK Channels for Optical Wireless Communication

A new optical steganography scheme is proposed that transmits a stealth optical code-division multiple-access (OCDMA) signal through a public binary phase-shift keying (BPSK) channel. Polarization beam splitters and arrayed waveguide gratings are used to implement a spectral-polarization coding (SPC) system with an incoherent optical source. We employ a Walsh–Hadamard code as the signature code of the user who wants to transmit stealth information using the system. A free space optical link applied to this system maintains the polarization states of light during propagation. The secret data are extracted using correlation detection and balanced subtraction in the OCDMA decoder of the intended receiver, and the other signal from the public channel is reduced by the OCDMA decoder. At the demodulator of the public channel, BPSK demodulation eliminates the stealth signal so that the public channel is not affected by the stealth signal. The two signals cannot interfere with each other. The results of this study show that our proposed optical steganography system is highly secure. The stealth signal can be favorably hidden in the public channel when the average source power of the stealth signal, public noise, and public signal are −5, −3, and 0 dBm, respectively.

A modified topology achieved in OFDM/SAC-OCDMA-based multi-diagonal code for enhancing spectral efficiency

In this paper, a modified topology-based orthogonal frequency division multiplexing–spectral amplitude coding in optical code division multiple access (OFDM–SAC-OCDMA) has been introduced. Such analysis aims to explain the effect of spectral efficiency with respect to multi-diagonal (MD) code. The new proposed technique introduces a fusion system between the two-code keying scheme and the forward error correction (FEC). The introduced analysis has targeted the effect of signal-to-noise ratio, bit error rate, and spectral efficiency with the presence of beat noise, phase incoherent intensity noise, and thermal noise. Numerical simulations have been utilized and tested in order to illustrate the validation of the (MD) code method to eliminate dominant noise. The calculation results show the following; the SNR and BER for MD under the presence of the previous discussed signal degradation is equal 161 and 1.2 × 10−10, respectively, according to a predefined number of users 150 comparing with some existing codes such as: modified frequency hopping code, Hadamard (HD) code, modified quadratic congruence code. The proposed unipolar encoding with direct detection technique-based OFDM/SAC-OCDMA scheme achieves an enhancement in the performance of SE over the unipolar encoding with direct detection technique-based SAC-OCDMA technique by 3.455 dB. The proposed two keying scheme-based OFDM/SAC-OCDMA introduces an enhancement in the performance of the SE over the unipolar encoding with direct detection technique-based OFDM/SAC-OCDMA technique by 2.8859 dB. FEC has been applied in order to increase the spectral efficiency for different BERs, which the numerical results show that FEC provides 1.7457 dB. Finally, the total enhancement for SE was about 8.0866 dB.

On $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard codes: kernel and partial classification

The $$\mathbb {Z}_{2^s}$$ -additive codes are subgroups of $$\mathbb {Z}^n_{2^s}$$ , and can be seen as a generalization of linear codes over $$\mathbb {Z}_2$$ and $$\mathbb {Z}_4$$ . A $$\mathbb {Z}_{2^s}$$ -linear Hadamard code is a binary Hadamard code which is the Gray map image of a $$\mathbb {Z}_{2^s}$$ -additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $$\mathbb {Z}_4$$ -linear Hadamard codes. In this paper, the kernel of $$\mathbb {Z}_{2^s}$$ -linear Hadamard codes of length $$2^t$$ and its dimension are established for $$s > 2$$ . Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to $$t=11$$ for any $$s\ge 2$$ , by using also the rank.

Proposal of a combined CFR estimation method for practical bidirectional ANC-OFDM system in higher time-varying fading channel

In bidirectional analogue network coding based on orthogonal frequency division multiplexing (ANC-OFDM) system, two users can transfer their data to each other via the Relay during two timeslots. Each user can demodulate the other user’s data after removing the self-data with equalizing by employing the combined channel frequency response (CFR) estimated at each user in the 2nd timeslot. From this reason, the accurate channel estimation method for the combined CFR acts on the bit-error-rate (BER) performance of the bidirectional ANC-OFDM system. To satisfy the requirement, a combined CFR estimation with Chu code and a combined CFR estimation with Walsh code methods were proposed for bidirectional ANC-OFDM system which can achieve the higher channel estimation accuracy. However, their accuracy of channel estimation would be degraded a lot at the non-Nyquist rate in the practical bidirectional ANC-OFDM system. To solve this problem, this paper proposes a combined CFR estimation method for practical bidirectional ANC-OFDM system which can improve much higher the accuracy of channel estimation and get better BER performance. The silent features of the proposed method are to estimate the combined CFR by applying the maximum likelihood technique with the proposed special pilot subcarrier arrangement which can provide the accurate channel estimation for the combined CFR in the practical bidirectional ANC-OFDM system and to estimate the combined CFR over one OFDM frame by applying the cubic spline interpolation technique which can achieve the higher channel estimation accuracy in higher time-varying fading channel. In the performance evaluations, the normalized mean square error (MSE) as the channel estimation accuracy and BER performances are evaluated by using the computer simulations in higher time-varying fading channel at the non-Nyquist rate. The excellent normalized MSE and BER performances of the proposed method have been verified by the computer simulations in this paper.

Hadamard Coding for Supervised Discrete Hashing.

In this paper, we propose a learning-based supervised discrete hashing method. Binary hashing is widely used for large-scale image retrieval as well as video and document searches because the compact binary code representation is essential for data storage and reasonable for query searches using bit-operations. The recently proposed supervised discrete hashing (SDH) method efficiently solves mixed-integer programming problems by alternating optimization and the discrete cyclic coordinate descent (DCC) method. Based on some preliminary experiments, we show that the SDH method can be simplified without performance degradation. We analyze the simplified model and provide a mathematically exact solution thereof; we reveal that the exact binary code is provided by a "Hadamard matrix." Therefore, we named our method Hadamard codedsupervised discrete hashing (HC-SDH). In contrast to SDH, our model does not require an alternating optimization algorithm and does not depend on initial values. HC-SDH is also easier to implement than iterative quantization (ITQ). Experimental results involving a large-scale database show that Hadamard coding outperforms conventional SDH in terms of precision, recall, and computational time. On the large datasets SUN-397 and ImageNet, HC-SDH provides a superior mean average of precision (mAP) and top-accuracy compared to the conventional SDH methods with the same code length and FastHash. The training time of HC-SDH is 170 times faster than conventional SDH and the testing time including the encoding time is seven times faster than FastHash which encodes using a binary-tree.

Weak Flip Codes and their Optimality on the Binary Erasure Channel

This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise . The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes , which goes back to Shannon et al. and which was shown to achieve the error exponent with a fixed number of codewords $\mathsf {M}$ , can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the $r$ - wise Hamming distance is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum $r$ -wise Hamming distance satisfies a generalized $r$ - wise Plotkin bound . The $r$ -wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the $r$ -wise Plotkin bound with equality for all $r$ . In the second part of this paper, these insights are applied to a binary erasure channel with an arbitrary erasure probability 0 $\delta$ $r$ -wise Hamming distance structure of the code. For a number of codewords $\mathsf {M}$ satisfying $\mathsf {M}\le 4$ and an arbitrary finite blocklength $n$ , the globally optimal codes (in the sense of minimizing the average error probability) are found. For $\mathsf {M}$ = 5 or $\mathsf {M}$ = 6 and an arbitrary finite blocklength $n$ , the optimal codes are conjectured. For larger $\mathsf {M}$ , observations regarding the optimal design are presented, e.g., that good codes have a large $r$ -wise Hamming distance structure for all $r$ . Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes.

Design and Optimization of Differential Chaos Shift Keying Scheme With Code Index Modulation

Energy usage and data rate of differential chaos shift keying (DCSK) are both inefficient since half of the symbol duration is used to transmit reference. To improve the energy efficiency and data rate, we propose a new chaos modulation scheme which combines DCSK with code index modulation (CIM) referred to as CIM-DCSK. In the proposed scheme, a short reference signal is transmitted first. Then P repeated reference signals, which serve as information-bearing signals, are simultaneously modulated by an information bit and a Walsh code with a specific index which is determined by an extra information symbol. The analytical bit error rate (BER) performance of the proposed scheme is derived and verified by simulations. Further, we improve the performance of the proposed system by applying noise-reduction and optimizing power coefficients on the reference and the information-bearing signals of CIM-DCSK. Simulation and analytical results show that these optimization methods significantly boost the BER performances of the proposed schemes. Finally, the BER performances of the proposed systems are compared with that of the conventional DCSK system and the short reference DCSK (SR-DCSK) system. Numerical results show that the proposed schemes outperform both DCSK and SR-DCSK.

위성항법신호 적용을 위한 Walsh-Hadamard 코드 기반 직교변조 신호의 신호획득 성능

A signal acquisition scheme for a Walsh - Hadamard (WH) code-based orthogonal modulation signal is proposed for the application to the satellite navigation signals. The proposed scheme is a correlator that takes the advantage of the Fast Walsh - Hadamard Transform (FWHT) structure for the symbol demodulation of WH code. The proposed method’s signal acquisition performance is simulated in terms of false alarm probability and missing probability, and it is confirmed that a performance improvement of more than 1 dB can be achieved over the conventional LEX signal in the additive white Gaussian noise (AWGN) channel.

Performance enhancement of optical code-division multiple-access systems using transposed modified Walsh code

This paper presents the construction of unipolar transposed modified Walsh code (TMWC) and analysis of its performance in optical code-division multiple-access (OCDMA) systems. Specifically, the signal-to-noise ratio, bit error rate (BER), cardinality, and spectral efficiency were investigated. The theoretical analysis demonstrated that the wavelength-hopping time-spreading system using TMWC was robust against multiple-access interference and more spectrally efficient than systems using other existing OCDMA codes. In particular, the spectral efficiency was calculated to be 1.0370 when TMWC of weight 3 was employed. The BER and eye pattern for the designed TMWC were also successfully obtained using OptiSystem simulation software. The results indicate that the proposed code design is promising for enhancing network capacity.

Hadamard coding of time-of-flight neutron reflectogram at grazing incidence

At grazing incidence, neutrons are reflected by a flat smooth surface. Neutron reflectivity is known as a promising tool for precise and non-destructive analysis of buried layers and interfaces of thin films. On the other hand, studying inhomogeneous samples, which is quite often the case in practice with thin films, has been a big challenge, mainly because of the difficulty in obtaining a small neutron beam without significant intensity loss. The present research examines an alternative efficient way to provide spatial resolution in neutron reflectivity without the use of any micro beam and/or any high-resolution position sensitive neutron detector. It has been demonstrated for the first time that the reflectogram, which is a projection at grazing incidence along a neutron's propagation direction, can be obtained by use of a Hadamard coded mask and a conventional point detector such as a 3He detector. This suggests that neutron reflectivity imaging will be realized by collecting further reflectograms as a function of in-plane rotation angles.

Extensions of Hadamard Codes Defined on Rings

In this study, some special matrices are constructed by choosing certain elements from finite rings and certain codes are written by using these matrices. These codes are extended to a field. Moreover these codes are classified and more good codes are written.

Channel estimation in ACO-OFDM employing different transforms for VLC

The pronounced eminency of assuring simultaneous illumination and communication has driven Visible Light Communication (VLC) to gain significant ubiquity in recent times. This paper proposes comb type pilot arrangement based channel estimation for Asymmetrically Clipped Optical OFDM (ACO-OFDM) and different multicarrier transmission systems like Discrete Hartley Transform (DHT)-based ACO-OFDM and Fast-Walsh Hadamard transform (FWHT)-based Hadamard Coded Modulation (HCM) over dispersive VLC channel. Various channel estimation algorithms like Least Square (LS), Minimum Mean Square Error (MMSE), and Interpolation techniques namely Linear, Spline, and Low-Pass are evaluated and compared for the aforesaid systems employing different orders of constellation. Here, an elaborate mathematical analysis is accomplished and Cramer Rao Lower Bound (CRLB) is derived for the channel estimation error. Simulated results emphasize that, ACO-OFDM and DHT-based ACO-OFDM have improved Bit Error Rate (BER) performance than HCM at lower Signal to Noise Ratio (SNR), while at higher SNRs HCM dominates the former. Furthermore, the simulated results evidences that, in all multicarrier systems MMSE algorithm has reducible probability of error than LS because, at higher SNRs LS is more susceptible to noise and Spline Interpolation outperforms both LS and MMSE. The simulated results are validated analytically demonstrating good agreement.

High-SNR spectrum measurement based on Hadamard encoding and sparse reconstruction

The denoising capabilities of the H-matrix and cyclic S-matrix based on the sparse reconstruction, employed in the Pixel of Focal Plane Coded Visible Spectrometer for spectrum measurement are investigated, where the spectrum is sparse in a known basis. In the measurement process, the digital micromirror device plays an important role, which implements the Hadamard coding. In contrast with Hadamard transform spectrometry, based on the shift invariability, this spectrometer may have the advantage of a high efficiency. Simulations and experiments show that the nonlinear solution with a sparse reconstruction has a better signal-to-noise ratio than the linear solution and the H-matrix outperforms the cyclic S-matrix whether the reconstruction method is nonlinear or linear.

Full-Optical TOAD based Walsh-Hadamard code generation

Transmission of information takes place simultaneously over the same available channel bandwidth in Code Division Multiple Access (CDMA) technique. The spread spectrum (SS) technique is used in CDMA systems for transmission of information by employing spreading codes. A unique spreading code, acts as a signature code, is assigned for each individual user. The signal occupies a bandwidth much larger than the minimum necessary to send the information in SS modulation technique. A synchronized reception with the code at the receiver is applied for dispreading the information before the data recovery. From a long-term, Walsh-Hadamard codes have been employed as spread spectrum codes in CDMA communications due to their ease of generation. Walsh-Hadamard codes are absolutely orthogonal binary user codes which have so many favourite applications in synchronous multicarrier communications. However, the optical application of Walsh-Hadamard codes turns important for optical CDMA to solve the above problem. For this, the designs need the use of nonlinear optics capable of controlling multi-valued signals. To achieve this goal all-optical terahertz optical asymmetric Demultiplexer based design of Walsh-Hadamard codes is explored in this paper.

On two-weight $$\mathbb {Z}_{2^k}$$ Z 2 k -codes

We determine the possible homogeneous weights of regular projective two-weight codes over $$\mathbb {Z}_{2^k}$$ of length $$n>3$$ , with dual Krotov distance $$d^{\lozenge }$$ at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When $$k=2$$ , we characterize the parameters of such codes as those of the inverse Gray images of $$\mathbb {Z}_4$$ -linear Hadamard codes, which have been characterized by their types by several authors.