scholarly journals An infinite family of Hadamard matrices constructed from Paley type matrices

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 815-834
Author(s):  
Adda Farouk ◽  
Qing-Wen Wang
Keyword(s):  
Prime Number ◽  
Hadamard Matrix ◽  
Infinite Family ◽  
Hadamard Matrices

An n x n matrix whose entries are from the set {1,-1} is called a Hadamard matrix if HH? = nIn. The Hadamard conjecture states that if n is a multiple of four then there always exists Hadamard matrices of this order. But their construction remain unknown for many orders. In this paper we construct Hadamard matrices of order 2q(q + 1) from known Hadamard matrices of order 2(q + 1), where q is a power of a prime number congruent to 1 modulo 4. We show then two ways to construct them. This work is a continuation of U. Scarpis? in [7] and Dragomir-Z. Dokovic?s in [10].

scholarly journals Hadamard Matrices and Strongly Regular Graphs with the $3$-e.c. Adjacency Property

10.37236/1545 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Anthony Bonato ◽  
W. H. Holzmann ◽  
Hadi Kharaghani
Keyword(s):  
Hadamard Matrix ◽  
Infinite Family ◽  
Hadamard Matrices ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Element Subset ◽  
Strongly Regular ◽  
Almost All ◽  
Paley Graphs

A graph is $3$-e.c. if for every $3$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex not in $S$ which is joined to every vertex in $T$ and to no vertex in $S\setminus T$. Although almost all graphs are $3$-e.c., the only known examples of strongly regular $3$-e.c. graphs are Paley graphs with at least $29$ vertices. We construct a new infinite family of $3$-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.

scholarly journals An infinite family of Williamson matrices

1977 ◽  
Vol 24 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Edward Spence
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Infinite Family ◽  
Williamson Matrices

AbstractIn this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

scholarly journals Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

2019 ◽  
Vol 22 (5) ◽  
pp. 953-974
Author(s):  
Ángel del Río ◽  
Mariano Serrano
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Infinite Family ◽  
Solvable Groups ◽  
Integral Group ◽  
Rational Group ◽  
Torsion Unit ◽  
Zassenhaus Conjecture ◽  
A Finite Group ◽  
Torsion Units

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

scholarly journals Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Roots Of Unity ◽  
Galois Rings ◽  
Type Complex ◽  
Complex Hadamard Matrix

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

scholarly journals Vertex-primitive half-transitive graphs

1994 ◽  
Vol 57 (1) ◽  
pp. 113-124 ◽  
Author(s):  
D. E. Taylor ◽  
Ming-Yao Xu
Keyword(s):  
Prime Number ◽  
Infinite Family ◽  
Primitive Groups ◽  
Almost All ◽  
Transitive Graphs ◽  
Vertex Primitive

AbstractGiven an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p) acting on the cosets of a subgroup isomorphic to A5. In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.

scholarly journals Group properties of Hadamard matrices

1976 ◽  
Vol 21 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Design ◽  
Image Position ◽  
Square Matrix

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

Semi-automorphisms of Hadamard matrices

1975 ◽  
Vol 77 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Image Position

A Hadamard matrix Hn is an n by n matrix H = [hij], i, j = 1, …, n in which every entry hij is + 1 or − 1, such thatIt is well known that possible orders are n = 1, 2 and n = 4m. An automorphism α of H is given by a pair P, Q of monomial ± 1 matrices such thatHere P permutes and changes signs of rows, while Q acts similarly on columns.

Skew-Hadamard Matrices of the Goethals-Seidel Type

1975 ◽  
Vol 27 (3) ◽  
pp. 555-560 ◽  
Author(s):  
Edward Spence
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Projective Planes ◽  
Hadamard Matrices

1. Introduction. We prove, using a theorem of M. Hall on cyclic projective planes, that if g is a prime power such that either 1 + q + q2 is a prime congruent to 3, 5 or 7 (mod 8) or 3 + 2q + 2q2 is a prime power, then there exists a skew-Hadamard matrix of the Goethals-Seidel type of order 4(1 + q + q2). (A Hadamard matrix H is said to be of skew type if one of H + I, H — lis skew symmetric. ) If 1 + q + q2 is a prime congruent to 1 (mod 8), then a Hadamard matrix, not necessarily of skew type, of order 4(1 + q + q2) is constructed. The smallest new Hadamard matrix obtained has order 292.

Criteria for a Hadamard Matrix to be Skew-Equivalent

1976 ◽  
Vol 28 (6) ◽  
pp. 1216-1223 ◽  
Author(s):  
Judith Q. Longyear
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Permutation Matrix ◽  
Type I ◽  
Signed Permutation ◽  
Zero Entry

A matrix H of order n = 4t with all entries from the set ﹛1, —1﹜ is Hadamard if HHt = 4tI. The set of Hadamard matrices is . A matrix is of type I or is skew-Hadamard if H = S — I where St = —S (some authors also use H = S + I). The set of type I members is . A matrix P is a signed permutation matrix if each row and each column has exactly one non-zero entry, and that entry is from the set ﹛1, —1﹜.

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