scholarly journals A class of cyclic (v; k1, k2, k3; λ) difference families with v ≡ 3 (mod 4) a prime

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Dragomir Ž. Ðokovic ◽  
Ilias S. Kotsireas
Keyword(s):  
Difference Sets ◽  
Hadamard Matrices ◽  
Difference Families ◽  
First Time

AbstractWe construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.

scholarly journals Symmetric Hadamard Matrices of Orders 268, 412, 436 and 604

2018 ◽  
pp. 2-8 ◽  
Author(s):  
N. A. Balonin ◽  
D. Z. Ðokovic'
Keyword(s):  
Hash Function ◽  
Hadamard Matrices ◽  
Orbit Method ◽  
Practical Relevance ◽  
Video Information ◽  
The Past ◽  
Difference Families ◽  
Feasible Sets ◽  
On Line ◽  
First Time

Purpose.To investigate more fully, than what was done in the past, certain families of symmetric Hadamard matrices of small orders by using the so called propus construction.Methods.Orbit method for the search of three cyclic blocks to construct Hadamard matrices of propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function.Results. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters.Practical relevance.Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network “Internet” together with executable on-line algorithms. 

scholarly journals Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

10.37236/8753 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Venkata Raghu Tej Pantangi
Keyword(s):  
Hadamard Matrix ◽  
Abelian Groups ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Critical Group ◽  
The Other ◽  
Critical Groups ◽  
Difference Families ◽  
Skew Hadamard Difference Sets ◽  
Four Blocks

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.   

scholarly journals Cocyclic Hadamard matrices and difference sets

2000 ◽  
Vol 102 (1-2) ◽  
pp. 47-61 ◽  
Author(s):  
Warwick de Launey ◽  
D.L. Flannery ◽  
K.J. Horadam
Keyword(s):  
Difference Sets ◽  
Hadamard Matrices

scholarly journals Two-level Cretan matrices constructed using SBIBD

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
N. A. Balonin ◽  
Jennifer Seberry
Keyword(s):  
Incidence Matrix ◽  
Block Design ◽  
Difference Sets ◽  
Russian Literature ◽  
Incomplete Block ◽  
Balanced Incomplete Block Design ◽  
Orthogonal Matrices ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
First Time

AbstractTwo-level Cretan matrices are orthogonal matrices with two elements, x and y. At least one element per row and column is 1 and the other element has modulus ≤ 1. These have been studied in the Russian literature for applications in image processing and compression. Cretan matrices have been found by both mathematical and computational methods but this paper concentrates on mathematical solutions for the first time.We give, for the first time, families of Cretan matrices constructed using the incidence matrix of a symmetric balanced incomplete block design and Hadamard related difference sets.

Partial geometric difference sets and partial geometric difference families

2018 ◽  
Vol 341 (9) ◽  
pp. 2490-2498 ◽  
Author(s):  
Yanxun Chang ◽  
Fengzhao Cheng ◽  
Junling Zhou
Keyword(s):  
Difference Sets ◽  
Difference Families ◽  
Geometric Difference
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