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 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

A Skew Hadamard Matrix of Order 52

1969 ◽  
Vol 21 ◽  
pp. 1319-1322 ◽  
Author(s):  
D. Blatt ◽  
G. Szekeres
Keyword(s):  
Hadamard Matrix ◽  
Projective Planes ◽  
Identity Matrix ◽  
Finite Projective Planes ◽  
Image Position ◽  
Square Matrix

1. A Hadamard (H-) matrix H = (hij) of order n is an n × n square matrix satisfying the conditionsfor all i, j ≦ n. A skew H-matrix is an H-matrix of the formwhere I is the identity matrix and S’ the transpose of 5. In particular,Skew H-matrices have applications in the theory of finite projective planes (2) and tournaments (4), also in the construction of H-matrices of certain orders. For example, if there is a skew H-matrix of order n, then there is an H-matrix of order n(n – 1) (Williamson, see (1, p. 213)).

scholarly journals Complex Hadamard Matrices contained in a Bose–Mesner algebra

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Identity Matrix ◽  
Rational Map ◽  
Absolute Value ◽  
Complex Projective ◽  
Square Matrix

AbstractAcomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to C

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.

scholarly journals Generalizing Krawtchouk Polynomials Using Hadamard Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Peter S. Chami ◽  
Bernd Sing ◽  
Norris Sookoo
Keyword(s):  
Recurrence Relations ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Krawtchouk Polynomials ◽  
Generator Polynomial ◽  
Orthogonality Relations ◽  
Square Matrix

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica.

scholarly journals (v, k, λ) Configurations and Hadamard matrices

1970 ◽  
Vol 11 (3) ◽  
pp. 297-309 ◽  
Author(s):  
Jennifer Wallis
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Image Position

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems: Theorem 1. If there exist (i) a skew-Hadamard matrix H = U+I of order h, (ii)m-type matrices M = W+I and N = NT of order m, (iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying (a) XYT, YZT and ZXT all symmetric, and (b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

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.

scholarly journals Hadamard matrices of Williamson type

1976 ◽  
Vol 21 (4) ◽  
pp. 481-486 ◽  
Author(s):  
Albert Leon Whiteman
Keyword(s):  
Hadamard Matrices ◽  
Circulant Matrices ◽  
Identity Matrix ◽  
Image Position

AbstractLet p be a prime ≡ 1 (mod 4) and put v = p(p + 1)/2. It is proved in this paper that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order υ. This result is used to construct Hadamard matrices of order 4υ that are of the type originally prescribed by Williamson.

scholarly journals On Construction of Hadamard Rhotrix using a Special Type of Mn- Matrix

2019 ◽  
Vol 8 (9) ◽  
pp. 2339-2343
Keyword(s):  
Coding Theory ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Orthogonality Property ◽  
Wide Range ◽  
Unique Approach ◽  
Square Matrix

The Hadamard matrix H is a square matrix with all the entries +1’s or -1’s which satisfies the property HHT = n In. Rhotrix is a new concept for mathematical enrichment with much scope for research and has a wide range of applications in coding theory and cryptography. Mn–matrix is also a matrix with  1 entry, like the Hadamard matrix, but the orthogonality property is not satisfied. It is shown in this paper that Hadamard matrices and thereby Hadamard rhotrices can be constructed by using a special type of Mn-matrix, named N- matrix, which is a unique approach

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