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 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

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).

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 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

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

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.

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.

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