Search results for: optimal binary linear codes

Authors: Skezeer John B. Paz, Ederlina G. Nocon

Abstract:

Finding an optimal binary linear code is a central problem in coding theory. A binary linear code C = [n, k, d] is called optimal if there is no linear code with higher minimum distance d given the length n and the dimension k. There are bounds giving limits for the minimum distance d of a linear code of fixed length n and dimension k. The lower bound which can be taken by construction process tells that there is a known linear code having this minimum distance. The upper bound is given by theoretic results such as Griesmer bound. One way to find an optimal binary linear code is to make the lower bound of d equal to its higher bound. That is, to construct a binary linear code which achieves the highest possible value of its minimum distance d, given n and k. Some optimal binary linear codes were presented by Andries Brouwer in his published table on bounds of the minimum distance d of binary linear codes for 1 ≤ n ≤ 256 and k ≤ n. This was further improved by Markus Grassl by giving a detailed construction process for each code exhibiting the lower bound. In this paper, we construct new optimal binary linear codes by using some construction processes on existing binary linear codes. Particularly, we developed an algorithm applied to the codes already constructed to extend the list of optimal binary linear codes up to 257 ≤ n ≤ 300 for k ≤ 7.

Keywords: bounds of linear codes, Griesmer bound, construction of linear codes, optimal binary linear codes

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Authors: Ni Wenli, He Jing

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Repeat–Accumulate (RA) codes are subclass of LDPC codes with fast encoder structures. In this paper, we consider a nonbinary extension of binary LDPC codes over GF(q) and construct a non-binary RA code and a non-binary QC-LDPC code over GF(2^4), we construct non-binary RA codes with linear encoding method and non-binary QC-LDPC codes with algebraic constructions method. And the BER performance of RA and QC-LDPC codes over GF(q) are compared with BP decoding and by simulation over the Additive White Gaussian Noise (AWGN) channels.

Keywords: non-binary RA codes, QC-LDPC codes, performance comparison, BP algorithm

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Authors: Eusebio Jr. Lina, Ederlina Nocon

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Cyclic codes are a fundamental subclass of linear codes that enjoy a very interesting algebraic structure. The class of skew cyclic codes (or θ-cyclic codes) is a generalization of the notion of cyclic codes. This a very large class of linear codes which can be used to systematically search for codes with good properties. A linear code with complementary dual (LCD code) is a linear code C satisfying C ∩ C^⊥ = {0}. This subclass of linear codes provides an optimum linear coding solution for a two-user binary adder channel and plays an important role in countermeasures to passive and active side-channel analyses on embedded cryptosystems. This paper aims to identify LCD codes from the class of skew cyclic codes. Let F_q be a finite field of order q, and θ be an automorphism of F_q. Some conditions for a skew cyclic code to be LCD were given. To this end, the properties of a noncommutative skew polynomial ring F_q[x, θ] of automorphism type were revisited, and the algebraic structure of skew cyclic code using its skew polynomial representation was examined. Using the result that skew cyclic codes are left ideals of the ring F_q[x, θ]/〈x^n-1〉, a characterization of a skew cyclic LCD code of length n was derived. A necessary condition for a skew cyclic code to be LCD was also given.

Keywords: LCD cyclic codes, skew cyclic LCD codes, skew cyclic complementary dual codes, theta-cyclic codes with complementary duals

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Authors: Amin Saeidi

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Using a representation-theoretic approach and considering G to be a finite primitive permutation group of degree n, our aim is to determine linear codes of length n that admit G as a permutation automorphism group. We can show that in some cases, every binary linear code admitting G as a permutation automorphism group is a submodule of a permutation module defined by a primitive action of G. As an illustration of the method, we consider the sporadic simple group M₁₁ and the unitary group U(3,3). We also construct some point- and block-primitive 1-designs from the supports of some codewords of the codes in the discussion.

Keywords: linear code, permutation representation, support design, simple group

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Authors: Kassoul Ilyes

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The field of error correcting codes has been revolutionized by the introduction of iteratively decoded codes. Among these, LDPC codes are now a preferred solution thanks to their remarkable performance and low complexity. The binary version of LDPC codes showed even better performance, although it’s decoding introduced greater complexity. This thesis studies the performance of binary LDPC codes using simplified weighted decisions. Information is transported between a transmitter and a receiver by digital transmission systems, either by propagating over a radio channel or also by using a transmission medium such as the transmission line. The purpose of the transmission system is then to carry the information from the transmitter to the receiver as reliably as possible. These codes have not generated enough interest within the coding theory community. This forgetfulness will last until the introduction of Turbo-codes and the iterative principle. Then it was proposed to adopt Pearl's Belief Propagation (BP) algorithm for decoding these codes. Subsequently, Luby introduced irregular LDPC codes characterized by a parity check matrix. And finally, we study simplifications on binary LDPC codes. Thus, we propose a method to make the exact calculation of the APP simpler. This method leads to simplifying the implementation of the system.

Keywords: LDPC, parity check matrix, 5G, BER, SNR

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Authors: Alla Levina, Sergey Taranov

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The classical approach to the providing noise immunity and integrity of information that process in computing devices and communication channels is to use linear codes. Linear codes have fast and efficient algorithms of encoding and decoding information, but this codes concentrate their detect and correct abilities in certain error configurations. To protect against any configuration of errors at predetermined probability can robust codes. This is accomplished by the use of perfect nonlinear and almost perfect nonlinear functions to calculate the code redundancy. The paper presents the error-correcting coding scheme using biorthogonal wavelet transform. Wavelet transform applied in various fields of science. Some of the wavelet applications are cleaning of signal from noise, data compression, spectral analysis of the signal components. The article suggests methods for constructing linear codes based on wavelet decomposition. For developed constructions we build generator and check matrix that contain the scaling function coefficients of wavelet. Based on linear wavelet codes we develop robust codes that provide uniform protection against all errors. In article we propose two constructions of robust code. The first class of robust code is based on multiplicative inverse in finite field. In the second robust code construction the redundancy part is a cube of information part. Also, this paper investigates the characteristics of proposed robust and linear codes.

Keywords: robust code, linear code, wavelet decomposition, scaling function, error masking probability

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Authors: Jing Li, Xiuli Li

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This paper studies a special class of linear codes, called skew cyclic codes, over the ring R= F_q+uF_q+…+u^k-1F_q, where q is a prime power. A Gray map ɸ from R to F_q and a Gray map ɸ' from Rⁿ to Fⁿ_q are defined, as well as an automorphism Θ over R. It is proved that the images of skew cyclic codes over R under map ɸ' and Θ are cyclic codes over F_q, and they still keep the dual relation.

Keywords: skew cyclic code, gray map, automorphism, cyclic code

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Authors: Salem Ben Said

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In recent times the decimal number system to which we are accustomed has received serious competition from the binary number system. In this note, an approach is suggested to teaching and learning the binary number system using examples from the real world. More precisely, we will demonstrate the utility of the binary system in describing the optimal strategy to win the Chinese Nim game, and in telegraphy by decoding the hidden message on Perseverance’s Mars parachute written in the language of binary system. Finally, we will answer the question, “why do modern computers prefer the ternary number system instead of the binary system?”. All materials are provided in a format that is conductive to classroom presentation and discussion.

Keywords: binary number system, Nim game, telegraphy, computers prefer the ternary system

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Authors: Ibrahim Özbek, Erdoğan Mehmet Özkan

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In this study, we generalize (k,n) threshold secret sharing scheme on the study Ozbek and Siap to the codes over the ring Fq+ αFq. In this way, it is mentioned that the method obtained in that article can also be used on codes over rings, and new advantages to be obtained. The method of securely sharing the key in cryptography, which Shamir first systematized and Massey carried over to codes, became usable for all error-correcting codes. The firewall of this scheme is based on the hardness of the syndrome decoding problem. Also, an open study area is left for those working for other rings and code classes. All codes that correct errors with this method have been the working area of this method.

Keywords: secret sharing scheme, linear codes, algebra, finite rings

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Authors: Jaskarn Singh Bhullar, Divya Taneja, Manish Gupta, Rajesh Kumar Narula

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The existence conditions of dual containing constacyclic codes have opened a new path for finding quantum maximum distance separable (MDS) codes. Using these conditions parameters of length n=(q²+1)/2 quantum MDS codes were improved. A class of quantum MDS codes of length n=(q²+q+1)/h, where h>1 is an odd prime, have also been constructed having large minimum distance and these codes are new in the sense as these are not available in the literature.

Keywords: hermitian construction, constacyclic codes, cyclotomic cosets, quantum MDS codes, singleton bound

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Authors: Nikolai Krainiukov, Boris Melnikov

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We study the properties of maximal prefix codes. The codes have many applications in computer science, theory of formal languages, data processing and data classification. Practical application is based on the representation of the maximal prefix codes as a sequence of words in a specific order. Our approach to study uses finite state automata (so-called flower automata) for the representation of prefix codes. An important task is the decomposition of prefix codes into prime prefix codes (factors). We discuss the properties of such prefix code decompositions. A linear time algorithm is designed which find the prime decomposition. To verify the correctness of the proposed algorithms, we implemented a system computer algebra GAP.

Keywords: maximal prefix code, regular languages, flower automata, prefix code decomposing

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Authors: N. H. Z. Abidin, N. F. M. Mokhtar, S. S. A. Gani

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The influence of diffusion of the thermal or known as Soret effect in a heated Binary fluid model with Coriolis force is investigated theoretically. The linear stability analysis is used, and the eigenvalue is obtained using the Galerkin method. The impact of the Soret and Coriolis force on the onset of stationary convection in a system is analysed with respect to various Binary fluid parameters and presented graphically. It is found that an increase of the Soret values, destabilize the Binary fluid layer system. However, elevating the values of the Coriolis force helps to lag the onset of convection in a system.

Keywords: Benard convection, binary fluid, Coriolis, Soret

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Authors: Qasim M. Kriri

Abstract:

Few Saudi Arabia production companies face financial profit issues until this moment. This work presents a linear integer programming model that solves a production problem of a Saudi Food Company in Saudi Arabia. An optimal solution to the above-mentioned problem is a Linear Programming solution. In this regard, the main purpose of this project is to maximize profit. Linear Programming Technique has been used to derive the maximum profit from production of natural juice at Saudi Food Co. The operations of production of the company were formulated and optimal results are found out by using Lindo Software that employed Sensitivity Analysis and Parametric linear programming in order develop Linear Programming. In addition, the parameter values are increased, then the values of the objective function will be increased.

Keywords: parameter linear programming, objective function, sensitivity analysis, optimize profit

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Authors: Sie Long Kek, Wah June Leong, Kok Lay Teo

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Linear quadratic Gaussian model is a standard mathematical model for the stochastic optimal control problem. The combination of the linear quadratic estimation and the linear quadratic regulator allows the state estimation and the optimal control policy to be designed separately. This is known as the separation principle. In this paper, an efficient computational method is proposed to solve the linear quadratic Gaussian problem. In our approach, the Hamiltonian function is defined, and the necessary conditions are derived. In addition to this, the output error is defined and the least-square optimization problem is introduced. By determining the first-order necessary condition, the gradient of the sum squares of output error is established. On this point of view, the stochastic approximation approach is employed such that the optimal control policy is updated. Within a given tolerance, the iteration procedure would be stopped and the optimal solution of the linear-quadratic Gaussian problem is obtained. For illustration, an example of the linear-quadratic Gaussian problem is studied. The result shows the efficiency of the approach proposed. In conclusion, the applicability of the approach proposed for solving the linear quadratic Gaussian problem is highly demonstrated.

Keywords: iteration procedure, least squares solution, linear quadratic Gaussian, output error, stochastic approximation

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Authors: Rajib Mia, Badam Singh Kushvah

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In this present paper, we use the photogravitational version of the restricted three body problem (RTBP) in binary systems. In the photogravitational RTBP, an infinitesimal particle (dust grain) is moving under the gravitational attraction and radiation pressure from the two bigger primaries. The third particle does not affect the motion of two bigger primaries. The zero-velocity curves, zero-velocity surfaces and their projections on the plane are studied. We have used existing analytical method to solve the equations of motion. We have obtained the Lagrangian points in some binary stellar systems. It is found that mass reduction factor affects the Lagrangian points. The linear stability of Lagrangian points is studied and found that these points are unstable. Moreover, trajectories of the infinitesimal particle at the triangular points are studied.

Keywords: binary systems, Lagrangian points, linear stability, photogravitational RTBP, trajectories

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Authors: Meetty Tomy, Arxhana G Thosar

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This paper provides the implementation of optimal control for an armature-controlled DC motor. The selection of error weighted Matrix and control weighted matrix in order to implement optimal control theory for improving the dynamic behavior of DC motor is presented. The closed loop performance of Armature controlled DC motor with derived linear optimal controller is then evaluated for the transient operating condition (starting). The result obtained from MATLAB is compared with that of PID controller and simple closed loop response of the motor.

Keywords: optimal control, DC motor, performance index, MATLAB

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Authors: Jaswinder Singh

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In this paper, comparative analysis is carried out to study the performance of wavelength/time/space optical CDMA codes using two well-known formats; those are CSRZ and DPSK-RZ using RSoft’s OptSIM. The analysis is carried out under the real-like scenario considering the presence of various non-linear effects such as XPM, SPM, SRS, SBS and FWM. Fiber dispersion and the multiple access interference are also considered. The codes used in this analysis are 3-D wavelength/time/space codes. These are converted into 2-D wavelength-time codes so that their requirement of space couplers and fiber ribbons is eliminated. Under the conditions simulated, this is found that CSRZ performs better than DPSK-RZ for fiber-optic CDMA applications.

Keywords: Optical CDMA, Multiple access interference (MAI), CSRZ, DPSK-RZ

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Authors: M. Chumburidze, N. Salia, T. Namchevadze

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This work deal with case processing problems in business. The task of strategic credit requirements management of cases portfolio is discussed. The information model of credit requirements in a binary tree diagram is considered. The algorithms to solve issues of prioritizing clusters of cases in business have been investigated. An implementation of priority queues to support case management operations has been presented. The corresponding pseudo codes for the programming application have been constructed. The tools applied in this development are based on binary tree ordering algorithms, optimization theory, and business management methods.

Keywords: credit network, case portfolio, binary tree, priority queue, stack

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Authors: M. P. Nanda Kumar, K. Dheeraj

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The design of a feedback controller, so as to minimize a given performance criterion, for a general non-linear dynamical system is difficult; if not impossible. But for a large class of non-linear dynamical systems, the open loop control that minimizes a performance criterion can be obtained using calculus of variations and Pontryagin’s minimum principle. In this paper, the open loop optimal trajectories, that minimizes a given performance measure, is used to train the neural network whose inputs are state variables of non-linear dynamical systems and the open loop optimal control as the desired output. This trained neural network is used as the feedback controller. In other words, attempts are made here to solve the “inverse optimal control problem” by using the state and control trajectories that are optimal in an open loop sense.

Keywords: inverse optimal control, radial basis function, neural network, controller design

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Authors: Fehmi Bouguezzi

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This paper studies optimal licensing regimes in a linear Hotelling model where firms are located at the end points of the city and where the transportation cost is not linear but quadratic. We study for that a more general cost function and we try to compare the findings with the results of the linear cost. We find the same optimal licensing regimes. A per unit royalty is optimal when innovation is not drastic and no licensing is better when innovation is drastic. We also find that no licensing is always better than fixed fee licensing.

Keywords: Hotelling model, technology transfer, patent licensing, quadratic transportation cost

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Authors: Arun Moorthy

Abstract:

Quantum computing promises to provide algorithmic speedups for a number of tasks; however, similar to classical computing, effective error-correcting codes are needed. Current quantum computers require costly equipment to control each particle, so having fewer particles to control is ideal. Although traditional quantum computers are built using qubits (2-level systems), qudits (more than 2-levels) are appealing since they can have an equivalent computational space using fewer particles, meaning fewer particles need to be controlled. Currently, qudit quantum error-correction codes are available for different level qudit systems; however, these codes have sometimes overly specific constraints. When building a qudit system, it is important for researchers to have access to many codes to satisfy their requirements. This project addresses two methods to increase the number of quantum error correcting codes available to researchers. The first method is generating new codes for a given set of parameters. The second method is generating new error-correction codes by using existing codes as a starting point to generate codes for another level (i.e., a 5-level system code on a 2-level system). So, this project builds a website that researchers can use to generate new error-correction codes or codes based on existing codes.

Keywords: qudit, error correction, quantum, qubit

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Authors: Abdulaziz Alsadhan, Naveed Khan

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In recent years intrusions on computer network are the major security threat. Hence, it is important to impede such intrusions. The hindrance of such intrusions entirely relies on its detection, which is primary concern of any security tool like Intrusion Detection System (IDS). Therefore, it is imperative to accurately detect network attack. Numerous intrusion detection techniques are available but the main issue is their performance. The performance of IDS can be improved by increasing the accurate detection rate and reducing false positive. The existing intrusion detection techniques have the limitation of usage of raw data set for classification. The classifier may get jumble due to redundancy, which results incorrect classification. To minimize this problem, Principle Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Local Binary Pattern (LBP) can be applied to transform raw features into principle features space and select the features based on their sensitivity. Eigen values can be used to determine the sensitivity. To further classify, the selected features greedy search, back elimination, and Particle Swarm Optimization (PSO) can be used to obtain a subset of features with optimal sensitivity and highest discriminatory power. These optimal feature subset used to perform classification. For classification purpose, Support Vector Machine (SVM) and Multilayer Perceptron (MLP) used due to its proven ability in classification. The Knowledge Discovery and Data mining (KDD’99) cup dataset was considered as a benchmark for evaluating security detection mechanisms. The proposed approach can provide an optimal intrusion detection mechanism that outperforms the existing approaches and has the capability to minimize the number of features and maximize the detection rates.

Keywords: Particle Swarm Optimization (PSO), Principle Component Analysis (PCA), Linear Discriminant Analysis (LDA), Local Binary Pattern (LBP), Support Vector Machine (SVM), Multilayer Perceptron (MLP)

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Authors: Khelifi Mustapha, A. Moulay lakhdar, I. Elawady

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We will study the unequal error protection for VQ image. We have used the Reed Solomon (RS) Codes as Channel coding because they offer better performance in terms of channel error correction over a binary output channel. One such channel (binary input and output) should be considered if it is the case of the application layer, because it includes all the features of the layers located below and on the what it is usually not feasible to make changes.

Keywords: vector quantization, channel error correction, Reed-Solomon channel coding, application

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Authors: Osama Mohamed Elrajubi, Idris El-Feghi, Mohamed Abu Baker Saghayer

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This paper describes an identification of specific shapes within binary images using the morphological Hit-or-Miss Transform (HMT). Hit-or-Miss transform is a general binary morphological operation that can be used in searching of particular patterns of foreground and background pixels in an image. It is actually a basic operation of binary morphology since almost all other binary morphological operators are derived from it. The input of this method is a binary image and a structuring element (a template which will be searched in a binary image) while the output is another binary image. In this paper a modification of Hit-or-Miss transform has been proposed. The accuracy of algorithm is adjusted according to the similarity of the template and the sought template. The implementation of this method has been done by C language. The algorithm has been tested on several images and the results have shown that this new method can be used for similar shape detection.

Keywords: hit-or-miss operator transform, HMT, binary morphological operation, shape detection, binary images processing

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Authors: Yuchen Yang, Zhenming Wang, Jun Zhu, Ning Zhao

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An easy-to-implement and robust finite volume multi-resolution Weighted Essentially Non-Oscillatory (WENO) scheme is proposed on adaptive cartesian grids in this paper. Such a multi-resolution WENO scheme is combined with the ghost cell immersed boundary method (IBM) and wall-function technique to solve Navier-Stokes equations. Unlike the k-exact finite volume WENO schemes which involve large amounts of extra storage, repeatedly solving the matrix generated in a least-square method or the process of calculating optimal linear weights on adaptive cartesian grids, the present methodology only adds very small overhead and can be easily implemented in existing edge-based computational fluid dynamics (CFD) codes with minor modifications. Also, the linear weights of this adaptive finite volume multi-resolution WENO scheme can be any positive numbers on condition that their sum is one. It is a way of bypassing the calculation of the optimal linear weights and such a multi-resolution WENO scheme avoids dealing with the negative linear weights on adaptive cartesian grids. Some benchmark viscous problems are numerical solved to show the efficiency and good performance of this adaptive multi-resolution WENO scheme. Compared with a second-order edge-based method, the presented method can be implemented into an adaptive cartesian grid with slight modification for big Reynolds number problems.

Keywords: adaptive mesh refinement method, finite volume multi-resolution WENO scheme, immersed boundary method, wall-function technique.

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Authors: María Rosa Mucci

Abstract:

Tzvetan Todorov (1971) designs a model of narrative transformation where the plot is constituted by difference and resemblance. This binary opposition is a synthesis of a central figure within narrative discourse: metaphor. Narrative operates as a metaphor since it combines different actions through similarities within a common plot. However, it sounds paradoxical that metonymy and not metaphor should be the key figure within the narrative. It is a metonymy that keeps the movement of actions within the story through syntagmatic relations. By the same token, this articulation of verbs makes it possible for the reader to engage in a dynamic interaction with the text, responding to the plot and mediating meanings with the contradictory external world. As Roland Barthes (1957) points out, there are two codes that are irreversible within the process: the codes of actions and the codes of enigmas. Virginia Woolf constructs her plots through a process of symbolism; a scene is always enduring, not only because it stands for something else but also because it connotes it. The reader is forced to elaborate the meaning at a mythological level beyond the lines. In this research, we follow a qualitative content analysis to code language through the proairetic (actions) and hermeneutic (enigmas) codes in terms of Barthes. There are two novels in particular that engage the reader in this process of construction: Mrs Dalloway (1925) and To the Lighthouse (1927). The bridge from the first to the second brings memories of childhood, allowing for the discovery of these enigmas hidden between the lines. What survives? Who survives? It is the reader's task to unravel these codes and rethink this dialogue between plot and reader to contribute to the predominance of texts and the textuality of narratives.

Keywords: metonymy, code, metaphor, myth, textuality

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Authors: P. Shrivastava, A. Shukla, K. Verma, S. Rungta

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Parkinson's disease is a chronic neurological disorder that directly affects human gait. It leads to slowness of movement, causes muscle rigidity and tremors. Gait serve as a primary outcome measure for studies aiming at early recognition of disease. Using gait techniques, this paper implements efficient binary bat algorithm for an early detection of Parkinson's disease by selecting optimal features required for classification of affected patients from others. The data of 166 people, both fit and affected is collected and optimal feature selection is done using PSO and Bat algorithm. The reduced dataset is then classified using neural network. The experiments indicate that binary bat algorithm outperforms traditional PSO and genetic algorithm and gives a fairly good recognition rate even with the reduced dataset.

Keywords: parkinson, gait, feature selection, bat algorithm

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Authors: K. Y. Tseng, C. Y. Wu

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This paper employs a heuristic algorithm to solve engineering problems including truss structure optimization and optimal chiller loading (OCL) problems. Two different type algorithms, real-valued differential evolution (DE) and modified binary differential evolution (MBDE), are successfully integrated and then can obtain better performance in solving engineering problems. In order to demonstrate the performance of the proposed algorithm, this study adopts each one testing case of truss structure optimization and OCL problems to compare the results of other heuristic optimization methods. The result indicates that the proposed algorithm can obtain similar or better solution in comparing with previous studies.

Keywords: differential evolution, Truss structure optimization, optimal chiller loading, modified binary differential evolution

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Authors: Jamal Laaouine, Mohammed Elhassani Charkani

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Let p be a prime and let b be an integer. MDS b-symbol codes are a direct generalization of MDS codes. The γ-constacyclic codes of length pˢ over the finite commutative chain ring Fₚm [u]/ < u² > had been classified into four distinct types, where is a nonzero element of the field Fₚm. Let C₃ be a code of Type 3. In this paper, we obtain the b-symbol distance db(C₃) of the code C₃. Using this result, necessary and sufficient conditions under which C₃ is an MDS b-symbol code are given.

Keywords: constacyclic code, repeated-root code, maximum distance separable, MDS codes, b-symbol distance, finite chain rings

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Authors: Jorge Lus Nisperuza Toledo, Juan Pablo Rubio Ospina, Daniel Santiago Umana, Hector Alejandro Alvarez

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This work develops and implements computational codes to calculate the optimal launch trajectories for sending a probe from the earth to different planets of the Solar system, making use of trajectories of the Hohmann and No-Hohmann type and gravitational assistance in intermediate steps. Specifically, the orbital elements, the graphs and the dynamic simulations of the trajectories for sending a probe from the Earth towards the planets Mercury, Venus, Mars, Jupiter, and Saturn are obtained. A detailed study was made of the state vectors of the position and orbital velocity of the considered planets in order to determine the optimal trajectories of the probe. For this purpose, computer codes were developed and implemented to obtain the orbital elements of the Mariner 10 (Mercury), Magellan (Venus), Mars Global Surveyor (Mars) and Voyager 1 (Jupiter and Saturn) missions, as an exercise in corroborating the algorithms. This exercise gives validity to computational codes, allowing to find the orbital elements and the simulations of trajectories of three future interplanetary missions with specific launch windows.

Keywords: gravitational assistance, Hohmann’s trajectories, interplanetary mission, orbital elements

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