TWO-CIRCULANT HADAMARD MATRICES, WEIGHING MATRICES, AND RYSER'S CONJECTURE

2018 ◽  
Vol 3 (94) ◽  
pp. 2-9 ◽  
Author(s):  
Yu. N. Balonin ◽  
◽  
A. M. Sergeev ◽  
Keyword(s):  
Hadamard Matrices ◽  
Weighing Matrices

scholarly journals Short Amicable sets and Kharaghani type orthogonal designs

2001 ◽  
Vol 64 (3) ◽  
pp. 495-504 ◽  
Author(s):  
Christos Koukouvinos ◽  
Jennifer Seberry
Keyword(s):  
Hadamard Matrices ◽  
Weighing Matrices ◽  
Orthogonal Designs

Dedicated to Professor George SzekeresShort amicable sets were introduced recently and have many applications. The construction of short amicable sets has lead to the construction of many orthogonal designs, weighing matrices and Hadamard matrices. In this paper we give some constructions for short amicable sets as well as some multiplication theorems. We also present a table of the short amicable sets known to exist and we construct some infinite families of short amicable sets and orthogonal designs.

scholarly journals New symmetric designs from regular Hadamard matrices

10.37236/1339 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Hadamard Matrices ◽  
Weighing Matrices ◽  
Symmetric Designs ◽  
Parameter Values

For every positive integer $m$, we construct a symmetric $(v,k,\lambda )$-design with parameters $v={{h((2h-1)^{2m}-1)}\over{h-1}}$, $k=h(2h-1)^{2m-1}$, and $\lambda =h(h-1)(2h-1)^{2m-2}$, where $h=\pm 3\cdot 2^d$ and $|2h-1|$ is a prime power. For $m\geq 2$ and $d\geq 1$, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order $9\cdot 4^d$.

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 Determination of all possible orders of weight 16 circulant weighing matrices

2006 ◽  
Vol 12 (4) ◽  
pp. 498-538 ◽  
Author(s):  
K.T. Arasu ◽  
Ka Hin Leung ◽  
Siu Lun Ma ◽  
Ali Nabavi ◽  
D.K. Ray-Chaudhuri
Keyword(s):  
Weighing Matrices

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.

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