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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

A Comparative Study of Distribution System Parameter Estimation Methods

In this paper, we compare two parameter estimation methods for distribution systems: residual sensitivity analysis and state-vector augmentation with a Kalman filter. These two methods were originally proposed for transmission systems, and are still the most commonly used methods for parameter estimation. Distribution systems have much lower measurement redundancy than transmission systems. Therefore, estimating parameters is much more difficult. To increase the robustness of parameter estimation, the two methods are applied with combined measurement snapshots (measurement sets taken at different points in time), so that the redundancy for computing the parameter values is increased. The advantages and disadvantages of both methods are discussed. The results of this paper show that state-vector augmentation is a better approach for parameter estimation in distribution systems. Simulation studies are done on a modified version of IEEE 13-Node Test Feeder with varying levels of measurement noise and non-zero error in the other system model parameters.

distribution system analysis↗
DOE OSTI2012

Broadband microwave spectroscopy in Corbino geometry at {sup 3}He temperatures

A broadband microwave spectrometer has been constructed to determine the complex conductivity of thin metal films at frequencies from 45 MHz to 20 GHz working in the temperature range from 0.45 K to 2 K (in a {sup 3}He cryostat). The setup follows the Corbino approach: a vector network analyzer measures the complex reflection coefficient of a microwave signal hitting the sample as termination of a coaxial transmission line. As the calibration of the setup limits the achievable resolution, we discuss the sources of error hampering different types of calibration. Test measurements of the complex conductivity of a heavy-fermion material demonstrate the applicability of the calibration procedures.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Machine Learning-Based Identification of the Interface Regions for Coupling Local and Nonlocal Models

Local-nonlocal coupling approaches provide a means to combine the computational efficiency of local models and the accuracy of nonlocal models. However, the coupling process can be challenging, requiring expertise to identify the interface between local and nonlocal regions. Here, this study introduces a machine learning-based approach to automatically detect the regions in which the local and nonlocal models should be used. The method uses loading functions evaluated at grid points to decide the model selection at those points. Training of the networks is based on datasets provided by classes of loading functions for which reference coupling configurations are computed using accurate coupled solutions, where accuracy is measured in terms of the relative error between the solution to the coupling approach and the solution to the nonlocal model. We study two approaches that vary in data structure. The first, the full-domain input data approach, uses the entire load vector and outputs a complete label vector, performing a global classification. The second, a window-based approach, processes loads into windows and addresses the problem as a node-wise classification where each window's central point is classified individually. The classification problems are solved via deep learning algorithms based on convolutional neural networks. The performance of these approaches is studied on one-dimensional numerical examples using F1-scores and accuracy metrics. Notably, the windowing approach achieves an accuracy of 0.96 and an F1-score of 0.97, highlighting its potential to automate coupling processes effectively and enhance computational efficiency in material science applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Model predictive control and optimization for battery charging and discharging

An apparatus for model predictive control (“MPC”) is disclosed. A method and system also perform the functions of the apparatus. The apparatus includes a measurement module that receives battery status information from one or more sensors receiving information from a battery cell, and a Kalman filter module that uses a Kalman filter and the battery status information to provide a state estimate vector. The apparatus includes a battery model module that inputs the state estimate vector and battery status information into a battery model and calculates a battery model output, the battery model representing the battery cell, and an MPC optimization module that inputs one or more battery model outputs and an error signal in a model predictive control algorithm to calculate an optimal response. The optimal response includes a modification of the error signal.

Trimboli, Michael Scott↗
DOE OSTI2012

D-Factor: A Quantitative Model of Application Slow-Down in Multi-Resource Shared Systems

Scheduling multiple jobs onto a platform enhances system utilization by sharing resources. The benefits from higher resource utilization include reduced cost to construct, operate, and maintain a system, which often include energy consumption. Maximizing these benefits comes at a price - resource contention among jobs increases job completion time. In this paper, we analyze slow-downs of jobs due to contention for multiple resources in a system; referred to as dilation factor. We observe that multiple-resource contention creates non-linear dilation factors of jobs. From this observation, we establish a general quantitative model for dilation factors of jobs in multi-resource systems. A job is characterized by a vector-valued loading statistics and dilation factors of a job set are given by a quadratic function of their loading vectors. We demonstrate how to systematically characterize a job, maintain the data structure to calculate the dilation factor (loading matrix), and calculate the dilation factor of each job. We validate the accuracy of the model with multiple processes running on a native Linux server, virtualized servers, and with multiple MapReduce workloads co-scheduled in a cluster. Evaluation with measured data shows that the D-factor model has an error margin of less than 16%. We also show that the model can be integrated with an existing on-line scheduler to minimize the makespan of workloads.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2015

Intrafraction Variability and Deformation Quantification in the Breast

Purpose: To evaluate intrafraction variability and deformation of the lumpectomy cavity (LC), breast, and nearby organs. Methods and Materials: Sixteen left-sided postlumpectomy and 1 bilateral breast cancer cases underwent free-breathing CT (FBCT) and 10-phase 4-dimensional CT (4DCT). Deformable image registration was used for deformation analysis and contour propagation of breast, heart, lungs, and LC between end-exhale and end-inhale 4DCT phases. Respiration-induced motion was calculated via centroid analysis. Two planning target volumes (PTVs) were compared: PTV{sub FBCT} from the FBCT volume with an isotropic 10 mm expansion (5 mm excursion and 5 mm setup error) and PTV{sub 4DCT} generated from the union of 4DCT contours with isotropic 5 mm margin for setup error. Volume and geometry were evaluated via percent difference and bounding box analysis, respectively. Deformation correlations between breast/cavity, breast/lung, and breast/heart were evaluated. Associations were tested between cavity deformation and proximity to chest wall and breast surface. Results: Population-based 3-dimensional vector excursions were 2.5 ± 1.0 mm (range, 0.8-3.8 mm) for the cavity and 2.0 ± 0.8 mm (range, 0.7-3.0 mm) for the ipsilateral breast. Cavity excursion was predominantly in the anterior and superior directions (1.0 ± 0.8 mm and −1.8 ± 1.2 mm, respectively). Similarly, for all cases, LCs and ipsilateral breasts yielded median deformation values in the superior direction. For 14 of 17 patients, the LCs and breast interquartile ranges tended toward the anterior direction. The PTV{sub FBCT} was 51.5% ± 10.8% larger (P<.01) than PTV{sub 4DCT}. Bounding box analysis revealed that PTV{sub FBCT} was 9.8 ± 1.2 (lateral), 9.0 ± 2.2 (anterior–posterior), and 3.9 ± 1.8 (superior–inferior) mm larger than PTV{sub 4DCT}. Significant associations between breast and cavity deformation were found for 6 of 9 axes. No dependency was found between cavity deformation and proximity to chest wall or breast surface. Conclusions: Lumpectomy cavity and breast deformation and motion demonstrated large variability. A PTV{sub 4DCT} approach showed value in patient-specific margins, particularly if robust interfraction setup analysis can be performed.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2020

Reconstruction of Charged Particle Tracks in Realistic Detector Geometry Using a Vectorized and Parallelized Kalman Filter Algorithm

One of the most computationally challenging problems expected for the High-Luminosity Large Hadron Collider (HL-LHC) is finding and fitting particle tracks during event reconstruction. Algorithms used at the LHC today rely on Kalman filtering, which builds physical trajectories incrementally while incorporating material e ects and error estimation. Recognizing the need for faster computational throughput, we have adapted Kalman-filterbased methods for highly parallel, many-core SIMD and SIMT architectures that are now prevalent in high-performance hardware. Previously we observed significant parallel speedups, with physics performance comparable to CMS standard tracking, on Intel Xeon, Intel Xeon Phi, and (to a limited extent) NVIDIA GPUs. While early tests were based on artificial events occurring inside an idealized barrel detector, we showed subsequently that our mkFit software builds tracks successfully from complex simulated events (including detector pileup) occurring inside a geometrically accurate representation of the CMS-2017 tracker. Here, we report on advances in both the computational and physics performance of mkFit, as well as progress toward integration with CMS production software. Recently we have improved the overall eciency of the algorithm by preserving short track candidates at a relatively early stage rather than attempting to extend them over many layers. Moreover, mkFit formerly produced an excess of duplicate tracks; these are now explicitly removed in an additional processing step. We demonstrate that with these enhancements, mkFit becomes a suitable choice for the first iteration of CMS tracking, and eventually for later iterations as well. We plan to test this capability in the CMS High Level Trigger during Run 3 of the LHC, with an ultimate goal of using it in both the CMS HLT and oine reconstruction for the HL-LHC CMS tracker.

Cerati, Giuseppe↗
DOE OSTI2019

fPINNs: Fractional Physics-Informed Neural Networks

Physics-informed neural networks (PINNs), introduced, are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE residuals and the mean-squared error in initial-boundary conditions is minimized with respect to the NN parameters. Here we extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we study systematically their convergence, hence explaining both fPINNs and PINNs for the first time. Specifically, we demonstrate their accuracy and effectiveness in solving multidimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box (BB) forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. This approach bypasses the difficulties stemming from the fact that automatic differentiation is not applicable to fractional operators because the standard chain rule in integer calculus is not valid in fractional calculus. To discretize the fractional operators, we employ the Grünwald--Letnikov (GL) formula in one-dimensional fractional ADEs and the vector GL formula in conjunction with the directional fractional Laplacian in two- and three-dimensional fractional ADEs. We first consider the one-dimensional fractional Poisson equation and compare the convergence of the fPINNs against the finite difference method (FDM). We present the solution convergence using both the mean L 2 error as well as the standard deviation due to sensitivity to NN parameter initializations. Using different GL formulas we observe first-, second-, and third-order convergence rates for small size training sets but the error saturates for larger training sets. We explain these results by analyzing the four sources of numerical errors due to discretization, sampling, NN approximation, and optimization. The total error decays monotonically (below 10 –5 for a third-order GL formula) but it saturates beyond that point due to the optimization error. We also analyze the relative balance between discretization and sampling errors and observe that the sampling size and the number of discretization points (auxiliary points) should be comparable to achieve the highest accuracy. As we increase the depth of the NN up to certain value, the mean error decreases and the standard deviation increases, whereas the width has essentially no effect unless its value is either too small or too large. We next consider time-dependent fractional ADEs and compare white-box (WB) and BB forcing. We observe that for the WB forcing, our results are similar to the aforementioned cases; however, for the BB forcing fPINNs outperform FDM. Subsequently, we consider multidimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of 10 –3 ~ 10 –4 . Lastly, we solve several inverse problems in one, two, and three dimensions to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results given proper initializations even in the presence of significant noise.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Distance-preserving rigidity penalty on deformable image registration of multiple skeletal components in the neck

Purpose: This study aims at developing and testing a novel rigidity penalty suitable for the deformable registration of tightly located skeletal components in the head and neck from planning computed tomography (CT) and daily cone-beam CT (CBCT) scans of patients undergoing radiotherapy. Methods: The proposed rigidity penalty is designed to preserve intervoxel distances within each bony structure. This penalty was tested in the intensity-based B-spline deformable registration of five cervical vertebral bodies (C1–C5). The displacement vector fields (DVFs) from the registrations were compared to the DVFs generated by using rigid body motions of the cervical vertebrae, measured by the surface registration of vertebrae delineated on CT and CBCT images. Twenty five pairs of planning CT (reference) and treatment CBCTs (target) from five patients were aligned without and with the penalty. An existing penalty based on the orthonormality of the deformation gradient tensor was also tested and the effects of the penalties compared. Results: The mean magnitude of the maximum registration error with the proposed distance-preserving penalty was (0.86, 1.12, 1.33) mm compared to (2.11, 2.49, 2.46) without penalty and (1.53, 1.64, 1.64) with the existing orthonormality-based penalty. The improvement in the accuracy of the deformable image registration was also verified by comparing the Procrustes distance between the DVFs. With the proposed penalty, the average distance was 0.11 (σ 0.03 mm) which is smaller than 0.53 (0.1 mm) without penalty and 0.28 (0.04 mm) with the orthonormality-based penalty. Conclusions: The accuracy of aligning multiple bony elements was improved by using the proposed distance-preserving rigidity penalty. The voxel-based statistical analysis of the registration error shows that the proposed penalty improved the integrity of the DVFs within the vertebral bodies.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Thermal neutron diffraction determination of the magnetic structure of EuCu{sub 2}Ge{sub 2}

The magnetic structure of EuCu{sub 2}Ge{sub 2} has been determined by flat-plate neutron powder diffraction. Two magnetic phases are present in the neutron diffraction pattern at 3.5 K. They have the same moment, within error, and a common transition temperature. Both {sup 151}Eu and {sup 153}Eu Mössbauer spectroscopy show that the two magnetic phases belong to the same crystallographic phase. Both phases can be modelled by planar helimagnetic structures: one with a propagation vector of [0.654(1), 0, 0], the other with a propagation vector of [0.410(1), 0.225(1), 0].

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2013

Interfractional Position Variation of Pancreatic Tumors Quantified Using Intratumoral Fiducial Markers and Daily Cone Beam Computed Tomography

Purpose: The aim of this study was to quantify interfractional pancreatic position variation using fiducial markers visible on daily cone beam computed tomography (CBCT) scans. In addition, we analyzed possible migration of the markers to investigate their suitability for tumor localization. Methods and Materials: For 13 pancreatic cancer patients with implanted Visicoil markers, CBCT scans were obtained before 17 to 25 fractions (300 CBCTs in total). Image registration with the reference CT was used to determine the displacement of the 2 to 3 markers relative to bony anatomy and to each other. We analyzed the distance between marker pairs as a function of time to identify marker registration error (SD of linear fit residuals) and possible marker migration. For each patient, we determined the mean displacement of markers relative to the reference CT (systematic position error) and the spread in displacements (random position error). From this, we calculated the group systematic error, Σ, and group random error, σ. Results: Marker pair distances showed slight trends with time (range, −0.14 to 0.14 mm/day), possibly due to tissue deformation, but no shifts that would indicate marker migration. The mean SD of the fit residuals was 0.8 mm. We found large interfractional position variations, with for 116 of 300 (39%) fractions a 3-dimensional vector displacement of >10 mm. The spread in displacement varied significantly (P<.01) between patients, from a vector range of 9.1 mm to one of 24.6 mm. For the patient group, Σ was 3.8, 6.6, and 3.5 mm; and σ was 3.6, 4.7 and 2.5 mm, in left–right, superior–inferior, and anterior–posterior directions, respectively. Conclusions: We found large systematic displacements of the fiducial markers relative to bony anatomy, in addition to wide distributions of displacement. These results for interfractional position variation confirm the potential benefit of using fiducial markers rather than bony anatomy for daily online position verification for pancreatic cancer patients.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Semiautomatic segmentation of aortic valve from sequenced ultrasound image using a novel shape-constraint GCV model

Purpose: Effective and accurate segmentation of the aortic valve (AV) from sequenced ultrasound (US) images remains a technical challenge because of intrinsic factors of ultrasound images that impact the quality and the continuous changes of shape and position of segmented objects. In this paper, a novel shape-constraint gradient Chan-Vese (GCV) model is proposed for segmenting the AV from time serial echocardiography. Methods: The GCV model is derived by incorporating the energy of the gradient vector flow into a CV model framework, where the gradient vector energy term is introduced by calculating the deviation angle between the inward normal force of the evolution contour and the gradient vector force. The flow force enlarges the capture range and enhances the blurred boundaries of objects. This is achieved by adding a circle-like contour (constructed using the AV structure region as a constraint shape) as an energy item to the GCV model through the shape comparison function. This shape-constrained energy can enhance the image constraint force by effectively connecting separate gaps of the object edge as well as driving the evolution contour to quickly approach the ideal object. Because of the slight movement of the AV in adjacent frames, the initial constraint shape is defined by users, with the other constraint shapes being derived from the segmentation results of adjacent sequence frames after morphological filtering. The AV is segmented from the US images by minimizing the proposed energy function. Results: To evaluate the performance of the proposed method, five assessment parameters were used to compare it with manual delineations performed by radiologists (gold standards). Three hundred and fifteen images acquired from nine groups were analyzed in the experiment. The area-metric overlap error rate was 6.89% ± 2.88%, the relative area difference rate 3.94% ± 2.63%, the average symmetric contour distance 1.08 ± 0.43 mm, the root mean square symmetric contour distance 1.37 ± 0.52 mm, and the maximum symmetric contour distance was 3.57 ± 1.72 mm. Conclusions: Compared with the CV model, as a result of the combination of the gradient vector and neighborhood shape information, this semiautomatic segmentation method significantly improves the accuracy and robustness of AV segmentation, making it feasible for improved segmentation of aortic valves from US images that have fuzzy boundaries.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2020

Deep learning surrogate model for kinetic Landau-fluid closure with collision

In this work, the kinetic Landau-fluid (LF) closure with collision and periodic boundary condition is used in the development of the deep learning (DL) surrogate model. A classical neural network, namely, feedforward neural network or sometimes termed multilayer perceptron, is constructed and trained to learn the kinetic LF closure in the static limit and arbitrary mean free path in configuration space. The preliminary relation between best hyperparameters and critical parameters for data generation is found. Compared with the numerical approach (non-Fourier method) of the LF closure, the deep learning surrogate model shows an order of magnitude of improvement in terms of accuracy. Perhaps most importantly, the surrogate model closure has been integrated for the first time with fluid simulations. Our DL-enabled fluid simulations, for the first time, give the correct Landau damping rate for a wide range of wave vectors, while the Hammett–Perkins closure cannot produce the correct damping rate. We correctly connect the collisionless Hammett–Perkins closure and collisional Braginskii closure to reproduce the intrinsic nonlocal feature of the heat flux with DL techniques. We address the most concerning error accumulation problem and find that simulations with the deep learning surrogate model are as good as, if not better than, simulations with the analytic closure in terms of long-term numerical stability in the linear Landau damping test.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

TESTING CPT SYMMETRY WITH CURRENT AND FUTURE CMB MEASUREMENTS

In this paper, we use the current and future cosmic microwave background (CMB) experiments to test the Charge-Parity-Time Reversal (CPT) symmetry. We consider a CPT-violating interaction in the photon sector L{sub cs}∼p{sub μ}A{sub ν} F-tilde {sup μν}, which gives rise to a rotation of the polarization vectors of the propagating CMB photons. By combining the 9 yr WMAP, BOOMERanG 2003, and BICEP1 observations, we obtain the current constraint on the isotropic rotation angle α-bar =−2.12±1.14 (1σ), indicating that the significance of the CPT violation is about 2σ. Here, we particularly take the systematic errors of CMB measurements into account. Then, we study the effects of the anisotropies of the rotation angle [Δα( n-hat )] on the CMB polarization power spectra in detail. Due to the small effects, the current CMB polarization data cannot constrain the related parameters very well. We obtain the 95% C.L. upper limit of the variance of the anisotropies of the rotation angle C {sup α}(0) < 0.035 from all of the CMB data sets. More interestingly, including the anisotropies of rotation angle could lower the best-fit value of r and relax the tension on the constraints of r between BICEP2 and Planck. Finally, we investigate the capabilities of future Planck polarization measurements on α-bar and Δα( n-hat ). Benefited from the high precision of Planck data, the constraints of the rotation angle can be significantly improved.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

TU-F-BRF-03: Effect of Radiation Therapy Planning Scan Registration On the Dose in Lung Cancer Patient CT Scans

Purpose: To characterize the effect of deformable registration of serial computed tomography (CT) scans on the radiation dose calculated from a treatment planning scan. Methods: Eighteen patients who received curative doses (≥60Gy, 2Gy/fraction) of photon radiation therapy for lung cancer treatment were retrospectively identified. For each patient, a diagnostic-quality pre-therapy (4–75 days) CT scan and a treatment planning scan with an associated dose map calculated in Pinnacle were collected. To establish baseline correspondence between scan pairs, a researcher manually identified anatomically corresponding landmark point pairs between the two scans. Pre-therapy scans were co-registered with planning scans (and associated dose maps) using the Plastimatch demons and Fraunhofer MEVIS deformable registration algorithms. Landmark points in each pretherapy scan were automatically mapped to the planning scan using the displacement vector field output from both registration algorithms. The absolute difference in planned dose (|ΔD|) between manually and automatically mapped landmark points was calculated. Using regression modeling, |ΔD| was modeled as a function of the distance between manually and automatically matched points (registration error, E), the dose standard deviation (SD-dose) in the eight-pixel neighborhood, and the registration algorithm used. Results: 52–92 landmark point pairs (median: 82) were identified in each patient's scans. Average |ΔD| across patients was 3.66Gy (range: 1.2–7.2Gy). |ΔD| was significantly reduced by 0.53Gy using Plastimatch demons compared with Fraunhofer MEVIS. |ΔD| increased significantly as a function of E (0.39Gy/mm) and SD-dose (2.23Gy/Gy). Conclusion: An average error of <4Gy in radiation dose was introduced when points were mapped between CT scan pairs using deformable registration. Dose differences following registration were significantly increased when the Fraunhofer MEVIS registration algorithm was used, spatial registration errors were larger, and dose gradient was higher (i.e., higher SD-dose). To our knowledge, this is the first study to directly compute dose errors following deformable registration of lung CT scans.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2013

Combined group ECC protection and subgroup parity protection

A method and system are disclosed for providing combined error code protection and subgroup parity protection for a given group of n bits. The method comprises the steps of identifying a number, m, of redundant bits for said error protection; and constructing a matrix P, wherein multiplying said given group of n bits with P produces m redundant error correction code (ECC) protection bits, and two columns of P provide parity protection for subgroups of said given group of n bits. In the preferred embodiment of the invention, the matrix P is constructed by generating permutations of m bit wide vectors with three or more, but an odd number of, elements with value one and the other elements with value zero; and assigning said vectors to rows of the matrix P.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Combined group ECC protection and subgroup parity protection

A method and system are disclosed for providing combined error code protection and subgroup parity protection for a given group of n bits. The method comprises the steps of identifying a number, m, of redundant bits for said error protection; and constructing a matrix P, wherein multiplying said given group of n bits with P produces m redundant error correction code (ECC) protection bits, and two columns of P provide parity protection for subgroups of said given group of n bits. In the preferred embodiment of the invention, the matrix P is constructed by generating permutations of m bit wide vectors with three or more, but an odd number of, elements with value one and the other elements with value zero; and assigning said vectors to rows of the matrix P.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Combined group ECC protection and subgroup parity protection

A method and system are disclosed for providing combined error code protection and subgroup parity protection for a given group of n bits. The method comprises the steps of identifying a number, m, of redundant bits for said error protection; and constructing a matrix P, wherein multiplying said given group of n bits with P produces m redundant error correction code (ECC) protection bits, and two columns of P provide parity protection for subgroups of said given group of n bits. In the preferred embodiment of the invention, the matrix P is constructed by generating permutations of m bit wide vectors with three or more, but an odd number of, elements with value one and the other elements with value zero; and assigning said vectors to rows of the matrix P.

97 MATHEMATICS AND COMPUTING↗