scholarly journals Analysis of implementations of the Scarpi method for calculating high orders Hadamard matrices of symmetric structures

2021 ◽  
Author(s):  
A.M. Sergeev ◽  
Keyword(s):  
Computational Experiment ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Structures ◽  
Matrix Calculation

An analysis of three modifications of the Scarpi method is given in order to assess their applicability to calculating Hadamard matrices of high orders with structural symmetries. Descriptions of modifications are presented, the results of Hadamard matrix calculation are demonstrated, confirming the conclusion about the significance of the Balonin-Seberry modification. The computational experiment shows that there are no results refuting the existence of matrices symmetric structures calculated by the Balonin-Seberry modification.

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

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﹜.

scholarly journals A skew Hadamard matrix of order 36

1970 ◽  
Vol 11 (3) ◽  
pp. 343-344 ◽  
Author(s):  
J. M. Goethals ◽  
J. J. Seidel
Keyword(s):  
Present Note ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Open Case

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

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.

scholarly journals Eigenvalues of Circulant Matrices and a Conjecture of Ryser

2021 ◽  
Vol 45 (5) ◽  
pp. 751-759
Author(s):  
REINHARDT EULER ◽  
◽  
LUIS H. GALLARDO ◽  
OLIVIER RAHAVANDRAINY
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Circulant Matrices ◽  
Algebraic Integers ◽  
Arithmetic Means ◽  
Complex Arithmetic

We prove that there is no circulant Hadamard matrix H with first row [h1,…,hn] of order n > 4, under some linear conditions on the hi’s. All these conditions hold in the known case n = 4, so that our results can be thought as characterizations of properties that only hold when n = 4. Our first conditions imply that some eigenvalue λ of H is a sum of √ -- n terms h jωj, where ω is a primitive n-th root of 1. The same conclusion holds also if some complex arithmetic means associated to λ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of robust Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very difficult to be able to prove the result.

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].

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