A practical algorithm for completing half-Hadamard matrices using LLL

2021 ◽  
Author(s):  
Assaf Goldberger ◽  
Yossi Strassler
Keyword(s):  
Hadamard Matrices ◽  
Practical Algorithm

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

scholarly journals A Practical Algorithm for General Large Scale Nonlinear Optimization Problems

1999 ◽  
Vol 9 (3) ◽  
pp. 755-778 ◽  
Author(s):  
Paul T. Boggs ◽  
Anthony J. Kearsley ◽  
Jon W. Tolle
Keyword(s):  
Nonlinear Optimization ◽  
Large Scale ◽  
Optimization Problems ◽  
Practical Algorithm

scholarly journals Practical algorithm to inform clinical decision‐making in the topical treatment of atopic dermatitis

2021 ◽  
Author(s):  
Thomas Luger ◽  
Uladzimir Adaskevich ◽  
Maryna Anfilova ◽  
Xia Dou ◽  
Nikolay N. Murashkin ◽  
...  
Keyword(s):  
Decision Making ◽  
Atopic Dermatitis ◽  
Topical Treatment ◽  
Clinical Decision Making ◽  
Clinical Decision ◽  
Practical Algorithm

scholarly journals Hadamard Matrices with Cocyclic Core

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 857
Author(s):  
Víctor Álvarez ◽  
José Andrés Armario ◽  
María Dolores Frau ◽  
Félix Gudiel ◽  
María Belén Güemes ◽  
...  
Keyword(s):  
Hadamard Matrices ◽  
Combinatorial Designs ◽  
Powerful Technique ◽  
Uniform Approach ◽  
Structured Approach

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture.

scholarly journals Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113

2018 ◽  
Vol 6 (1) ◽  
pp. 11-22 ◽  
Author(s):  
N. A. Balonin ◽  
D. Ž. Ðokovic ◽  
D. A. Karbovskiy
Keyword(s):  
Hadamard Matrices ◽  
Systematic Search

Abstract We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders 4v with odd v ≤ 41. In this paper we cover the cases v = 43, 45, 47, 49, 51. The odd integers v < 120 for which no symmetric Hadamard matrices of order 4v are known are the following: 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109, 113, 119. By using the propus construction, we found several symmetric Hadamard matrices of order 4v for v = 47, 73, 113.

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