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.

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 $9\cdot 4^d$.

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.

scholarly journals A new class of Hadamard matrices

1967 ◽  
Vol 8 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. Spence
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Identity Matrix ◽  
New Class ◽  
Image Position ◽  
Square Matrix

A Hadamard matrixHis an orthogonal square matrix of ordermall the entries of which are either + 1 or - 1; i. e.whereH′denotes the transpose ofHandImis the identity matrix of orderm. For such a matrix to exist it is necessary [1] thatIt has been conjectured, but not yet proved, that this condition is also sufficient. However, many values ofmhave been found for which a Hadamard matrix of ordermcan be constructed. The following is a list of suchm(pdenotes an odd prime).

scholarly journals Applying Balanced Generalized Weighing Matrices to Construct Block Designs

10.37236/1556 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Block Designs ◽  
Weighing Matrices ◽  
Symmetric Designs

Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters $(1+qr(r^{m+1}-1)/(r-1),r^{m},r^{m-1}(r-1)/q)$, where $m$ is any positive integer and $q$ and $r=(q^{d}-1)/(q-1)$ are prime powers, and a family of non-embeddable quasi-residual $2-((r+1)(r^{m+1}-1)/(r-1),r^{m}(r+1)/2,r^{m}(r-1)/2)$ designs, where $m$ is any positive integer and $r=2^{d}-1$, $3\cdot 2^{d}-1$ or $5\cdot 2^{d}-1$ is a prime power, $r\geq 11$.

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 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.

Modular Hadamard Matrices and Related Designs, II

1972 ◽  
Vol 24 (6) ◽  
pp. 1100-1109 ◽  
Author(s):  
O. Marrero ◽  
A. T. Butson
Keyword(s):  
Positive Integer ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Inner Product ◽  
Combinatorial Designs ◽  
Definition Of ◽  
Fixed Positive Integer

Anhbyhmatrix with entries ±1 is called amodular Hadamard matrixif the inner product of any two distinct row vectors is a multiple of a fixed (positive) integern;such a matrix is also referred to as an“H(n, h) matrix” with parametersnandh.Modular Hadamard matrices and the related combinatorial designs were introduced in [2]; that paper was concerned mainly with two of the related designs, the “pseudo (ν, k, λ)- designs” and the “ (m, v, k1,λ1,k2,λ2,f, λ3)-designs” (the reader is referred to [2] for the definition of these 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.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

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