The Densest Packing of 9 Circles in a Square

1965 ◽  
Vol 8 (3) ◽  
pp. 273-277 ◽  
Author(s):  
J. Schaer
Keyword(s):  
Minimum Distance ◽  
Packing Problems ◽  
Unit Square ◽  
Minimum Distances

Packing problems of this kind are obviously equivalent to the problems of placing k (here 9) points in a unit square such that the minimum distance between any two of them be as large as possible. The solutions of these problems are known for 2 ≤ k ≤ 9. The largest possible minimum distances mk are given in table 1, and the corresponding "best" configurations shown in figure 1.

scholarly journals A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

2019 ◽  
Vol 9 (2) ◽  
pp. 1232
Author(s):  
Issam Abderrahman Joundan ◽  
Said Nouh ◽  
Mohamed Azouazi ◽  
Abdelwahed Namir
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Search Problem ◽  
Public Key Cryptosystems ◽  
True Value ◽  
Efficient Scheme ◽  
Minimum Distances

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity.  ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

scholarly journals Determination of the critical distance in the procedure of explosive welding

2020 ◽  
Vol 68 (4) ◽  
pp. 823-844
Author(s):  
Miloš Lazarević ◽  
Bogdan Nedić ◽  
Jovica Bogdanov ◽  
Stefan Đurić
Keyword(s):  
Hearing Loss ◽  
Explosive Welding ◽  
Minimum Distance ◽  
Critical Distance ◽  
Maximum Pressure ◽  
Limit Pressure ◽  
Minimum Distances ◽  
Wave Methods ◽  
Welding Procedure

Introduction/purpose: When performing the explosive welding procedure, for the safety of workers, it is necessary to take into account the minimum distance between the workers and the place of explosion at the time of explosion. Negligence can cause temporary hearing loss, rupture of the eardrum and in some cases even the death of workers. The aim of this paper is to determine the critical distance based on the mass of explosive charge required for explosive welding, provided that the limit pressure is 6.9 kPa in the case of temporary hearing loss and 35 kPa in the case of eardrum rupture. This paper does not take into account other effects of the explosion than those caused by the shock wave. Methods: Depending on the type of explosion, the equivalent explosive mass was calculated. Based on the equivalent explosive mass and the limit pressure, the minimum distances were calculated using the Sadovsky and Kingery-Bulmash equations. Results: The corresponding tables show the results of the calculation of the critical distance of workers from the place of the explosion when there may be temporary hearing loss or rupture of the eardrum. The calculated value of the critical explosion distance by the Kingery-Bulmash method, under the condition of the maximum pressure for temporary hearing loss, is 5.62% lower than the distance value obtained by the Sadovsky method while the value of the critical explosion distance calculated by the Kingery-Bulmash method, under the condition of the maximum pressure for eardrum rupture, is 7.83% lower than the value obtained by the Sadovsky method. Conclusion: The results of the calculation showed that the critical distance from the explosion can be successfully calculated and that the obtained values have small differences depending on the applied calculation method.

scholarly journals Assessment of the capabilities of orbital optical devices for obtaining information about space objects

2021 ◽  
Vol 20 (2) ◽  
pp. 270-301
Author(s):  
Valeriy Prorok ◽  
Anatoliy Karytko ◽  
Alexander Goryanskiy ◽  
Ekaterina Emelyanova
Keyword(s):  
Minimum Distance ◽  
Fuzzy Inference ◽  
Optical Devices ◽  
Time Interval ◽  
Optimal Conditions ◽  
Relative Speed ◽  
Space Objects ◽  
Research Problems ◽  
Minimum Distances ◽  
Minimum Points

The purpose of the study is to select the optimal conditions for collecting non-coordinate information about a spacecraft by a space optical-electronic means at the time objects pass the vicinity of the points of the minimum distance between their orbits. The quantitative indicator is proposed that characterize the measure of the possibility of obtaining non-coordinate information about space objects with the required level of quality. The arguments of the function characterizing the indicator are the distance between spacecraft; their relative speed; phase angle of illumination of a spacecraft by the Sun in relation to the optical-electronic means; the length of the time interval during which both objects are in the vicinity of the point of a minimum distance between their orbits. The value of the indicator is computed by solving three particular research problems. The first task is to search for neighborhoods that include the minimum distances between the orbits of the controlled spacecraft and optical-electronic means. To solve it, a fast algorithm for calculating the minimum distance between orbits used. Additionally, the drift of the found neighborhoods is taken into account on the time interval up to 60 hours. The second task is to estimate the characteristics of motion and the conditions of optical visibility of a controlled spacecraft in the vicinity of the minimum points of the distance between the orbits of spacecraft. The solution to this problem is carried out by using the SGP4 library of space objects motion forecast. The third task is justification and calculation of an index characterizing the measure of the possibility of obtaining an optical image of a spacecraft for given conditions of optical visibility. To solve the problem, the developed system of fuzzy inference rules and the Mamdani algorithm is used. The presented method is implemented as a program. In the course of a computational experiment, an assessment was made of the possibility of obtaining non-coordinate information on low-orbit and geostationary space objects. The proposed indicator provides an increase in the efficiency of the procedure for collecting non-coordinate information about space objects by choosing the most informative alternatives for monitoring space objects from the available set of possible observations at a given planning interval for collecting information about space objects.

scholarly journals Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 78
Author(s):  
Lucian Trifina ◽  
Daniela Tarniceriu ◽  
Jonghoon Ryu ◽  
Ana-Mirela Rotopanescu
Keyword(s):  
Upper Bound ◽  
Turbo Codes ◽  
Minimum Distance ◽  
Prime Numbers ◽  
Upper Bounds ◽  
The Other ◽  
Small Range ◽  
Generator Matrix ◽  
Fourth Degree ◽  
Minimum Distances

In this paper, we obtain upper bounds on the minimum distance for turbo codes using fourth degree permutation polynomial (4-PP) interleavers of a specific interleaver length and classical turbo codes of nominal 1/3 coding rate, with two recursive systematic convolutional component codes with generator matrix G = [ 1 , 15 / 13 ] . The interleaver lengths are of the form 16 Ψ or 48 Ψ , where Ψ is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime p i ∣ Ψ , condition 3 ∤ ( p i − 1 ) is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP f 4 x 4 + f 3 x 3 + f 2 x 2 + f 1 x ( mod 16 k L Ψ ) , k L ∈ { 1 , 3 } , the upper bound of 28 is obtained when the coefficient f 3 of the equivalent 4-permutation polynomials (PPs) fulfills f 3 ∈ { 0 , 4 Ψ } or when f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 4 k L − 1 ) · Ψ , ( 8 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient f 3 of the equivalent 4-PPs fulfills f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 2 k L − 1 ) · Ψ , ( 6 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. Thus, the task of finding out good 4-PP interleavers of the previous mentioned lengths is highly facilitated by this result because of the small range required for coefficients f 4 , f 3 and f 2 . It was also proven, by means of nonlinearity degree, that for the considered inteleaver lengths, cubic PPs and quadratic PPs with optimum minimum distances lead to better error rate performances compared to 4-PPs with optimum minimum distances.

A comparison between Suzuki invariant code and one-point Hermitian code

2021 ◽  
pp. 2150104
Author(s):  
Abdulla Eid
Keyword(s):  
Algebraic Geometry ◽  
Minimum Distance ◽  
Algebraic Curves ◽  
Automorphism Groups ◽  
Algebraic Geometry Codes ◽  
Hermitian Codes ◽  
Relative Minimum ◽  
Minimum Distances ◽  
Hermitian Code

In this paper we compare the performance of two algebraic geometry codes (Suzuki and Hermitian codes) constructed using maximal algebraic curves over [Formula: see text] with large automorphism groups by choosing specific divisors. We discuss their parameters, compare the rate of these codes as well as their relative minimum distances, and we show that both codes are asymptotically good in terms of the rate which is in contrast to their behavior in terms of the relative minimum distance.

scholarly journals OPTIMIZED DETERMINATION OF 3D COORDINATES IN THE SURVEY OF INACCESSIBLE POINTS OF BUILDINGS - EXAMPLE OF APPLICATION IMPLEMENTED IN FREE SOFTWARE

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Leandro Luiz Silva de França ◽  
Andréa de Seixas ◽  
Luciene Ferreira Gama ◽  
João Naves de Moraes
Keyword(s):  
Minimum Distance ◽  
Geodetic Survey ◽  
Observation Point ◽  
Large Structures ◽  
Angle Measurements ◽  
Minimum Distances ◽  
Definition Of ◽  
Intersection Method ◽  
Better Than

Abstract: The forward intersection method is already widely used in the geodetic survey of coordinates of inaccessible points, especially when only angle measurements are available, in this case, also called the triangulation method. However, the mathematical solution of the 3D forward intersection with the analytical definition of spatial lines, resolved by the Minimum Distances Method, is still not widespread in the academic and professional environment. This mathematical modeling determines the 3D coordinates of a point located in the middle of the minimum distance between two or more spatial lines, which spatially "intersect" towards the observation point. This solution is more accurate than others presented in the literature because it simultaneously solves the problem of 3D determination of a point by the method of least squares, in addition to providing an estimate of the coordinate precision, which are inherent to the adjustment. This work, therefore, has the objective of explaining the Minimum Distances Method for the spatial intersection of targeted measurements with a Total Station from two or more known observation points for the 3D determination of inaccessible points located in corners of buildings. For the analysis of the method, a Python tool was developed for QGIS that calculates the 3D coordinates and generates the adjustment processing report, being applied with real observations of the Geodetic survey of the SUDENE building, in Recife-PE. The methodology developed in this work proved to be suitable for measurements of large structures, achieving spherical precision better than ±1.0 cm, following the Brazilian standards for urban cadastre.

scholarly journals List circular backbone colouring

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Frédéric Havet ◽  
Andrew King
Keyword(s):  
Chromatic Number ◽  
Minimum Distance ◽  
Natural Generalization ◽  
Graph Colouring ◽  
Combinatorial Nullstellensatz ◽  
International Audience ◽  
Individual Distance ◽  
Minimum Distances ◽  
List Chromatic Number ◽  
Circular Choosability

Graph Theory International audience A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.

New entanglement-assisted quantum MDS codes

2021 ◽  
pp. 2150016
Author(s):  
Binbin Pang ◽  
Shixin Zhu ◽  
Liqi Wang
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Maximum Distance Separable ◽  
Number Formula ◽  
Minimum Distances ◽  
Quantum Error ◽  
Quantum Mds Codes

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.

P35 Ability to quantify stereoelectroencephalography (SEEG) electrode trajectory proximity to vessels across imaging protocols

2019 ◽  
Vol 90 (3) ◽  
pp. e34.1-e34
Author(s):  
R Sparks ◽  
V Vakharia ◽  
R Rodionov ◽  
S Vos ◽  
A McEvoy ◽  
...  
Keyword(s):  
Phase Contrast ◽  
Minimum Distance ◽  
Mr Angiography ◽  
Automated Planning ◽  
Vessel Segmentation ◽  
Mr Venography ◽  
Vessel Size ◽  
Vertebral Arteries ◽  
Imaging Protocols ◽  
Minimum Distances

ObjectivesAutomated planning of stereoelectroencephalography (SEEG) electrode trajectories is dependent on vessel segmentation.1 We quantify imaging protocols ability to measure trajectory-to-vessel distance.DesignRetrospective analysis.SubjectsTen consecutive patients were selected whom had SEEG implantation (95 electrodes) and Digital Catheter Subtraction Angiography (DSA) with catheterization of carotid or vertebral arteries, post-gadolinium T1-weighted (GAD), phase-contrast MR angiography and MR venography (MR) acquired.MethodsSEEG trajectories were planned manually with DSA. Minimum distance to vessels and risk1 were computed for each trajectory using vessel segmentation from GAD, MR, or DSA. Vessel size was considered by including DSA vessels diameters above 1, 2, 3, or 4 mm.ResultsMinimum distance to a vessel was 6.2±3.9 mm (GAD), 2.5±1.6 mm (MR), and 1.5±1.2 mm (DSA). Based on DSA vessel size minimum distances were 2.0±1.5 mm (DSA >1 mm), 3.4±2.6 (DSA >2 mm), 6.6±4.6 mm (DSA >3 mm), and 11.8±7.9 mm (DSA >4 mm). Risk was 0.4±0.4 (GAD), 0.8±0.4 (MR), and 1.1±0.2 (all DSA), 1.0±0.2 (DSA >1 mm), 0.7±0.4 (DSA >2 mm), 0.4±0.5 (DSA >3 mm), and 0.2±0.3 (DSA >4 mm).ConclusionsDSA is best able to segment vessels. MR has metrics similar to DSA vessels above 2 mm. GAD has metrics similar to DSA vessels above 3 mm.

Reducing the risk of collision between ships in a close-quarters situation by simulating collision avoidance actions

2021 ◽  
pp. 1-16
Author(s):  
Djani Mohovic ◽  
Robert Mohovic ◽  
Marko Suljic ◽  
Marko Njegovan
Keyword(s):  
Collision Avoidance ◽  
Minimum Distance ◽  
International Regulations ◽  
Full Form ◽  
Minimum Distances ◽  
Definition Of

Abstract ‘Close-quarters situation’ is a term used in the International Regulations for Preventing Collisions at Sea. As the term is not precisely defined, this paper analyses the interpretations and definitions of the term by various authors or courts, based on judicial processes and judgments. In the end, the authors suggest their own definition of the term ‘close-quarters situation’. Knowing the minimum distance from another ship and the time to the closest point of approach at which collision may still be avoided by one's own manoeuvre is information that every ship's officer needs to know. In accordance with the proposed definition of the term ‘close-quarters situation’, minimum distances between ships and time to the closest point of approach in which the ship can still take action to avoid a collision by its own manoeuvring are determined by means of simulations on a navigational simulator. A total of 168 simulations were performed with three fine-form vessel sizes and three full-form vessel sizes. Due to the extensive amount of data, the paper presents the results for one vessel only.

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