scholarly journals Some new series of Hadamard matrices

1989 ◽  
Vol 46 (3) ◽  
pp. 371-383 ◽  
Author(s):  
Mieko Yamada
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Gauss Sums ◽  
Quaternion Type ◽  
Generalized Quaternion

AbstractThe purpose of this paper is to prove (1) if q ≡ 1 (mod 8) is a prime power and there exists a Hadamard matrix of order (q − 1)/2, then we can construct a Hadamard matrix of order 4q, (2) if q ≡ 5 (mod 8) is a prime power and there exists a skew-Hadamard matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2), (3) if q ≡ 1 (mod 8) is a prime power and there exists a symmetric C-matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2).We have 36, 36 and 8 new orders 4n for n ≤ 10000, of Hadamard matrices from the first, the second and third theorem respectively, which were known to the list of Geramita and Seberry. We prove these theorems by using an adaptation of generalized quaternion type array and relative Gauss sums.

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.

On C-Matrices of Arbitrary Powers

1971 ◽  
Vol 23 (3) ◽  
pp. 531-535 ◽  
Author(s):  
Richard J. Turyn
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Main Diagonal ◽  
Quadratic Character ◽  
Symmetric Complex ◽  
Complex Hadamard Matrix ◽  
Square Matrix

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

scholarly journals On the Twin Designs with the Ionin–type Parameters

10.37236/1479 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
H. Kharaghani
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Designs ◽  
New Class

Let $4n^2$ be the order of a Bush-type Hadamard matrix with $q=(2n-1)^2$ a prime power. It is shown that there is a weighing matrix $$ W(4(q^m+q^{m-1}+\cdots+q+1)n^2,4q^mn^2) $$ which includes two symmetric designs with the Ionin–type parameters $$ \nu=4(q^m+q^{m-1}+\cdots+q+1)n^2,\;\;\; \kappa=q^m(2n^2-n), \;\;\; \lambda=q^m(n^2-n) $$ for every positive integer $m$. Noting that Bush–type Hadamard matrices of order $16n^2$ exist for all $n$ for which an Hadamard matrix of order $4n$ exist, this provides a new class of symmetric designs.

New Families of Complex Hadamard Matrices

2015 ◽  
Vol 22 (03) ◽  
pp. 1550017 ◽  
Author(s):  
Maarten Havinga
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
The Real ◽  
Type Formula

The main result of this paper is a construction for complex Hadamard matrices: for [Formula: see text] any prime power and [Formula: see text] the size of a real Hadamard matrix, this construction yields a family of complex Hadamard matrices of order [Formula: see text] with [Formula: see text] parameters, including Butson-type matrices of even type [Formula: see text] a divisor of [Formula: see text]. Only a few lowdimensional examples and the real Hadamard matrices obtained by this construction are already known. Also a small extension of Diţa’s construction (cf. Lemma 1) is given.

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 Algebras of generalized quaternion type

2019 ◽  
Vol 349 ◽  
pp. 1036-1116 ◽  
Author(s):  
Karin Erdmann ◽  
Andrzej Skowroński
Keyword(s):  
Quaternion Type ◽  
Generalized Quaternion

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 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

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