scholarly journals Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

10.37236/4915 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Hadi Kharaghani ◽  
Sara Sasani ◽  
Sho Suda
Keyword(s):  
Upper Bound ◽  
Association Scheme ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Mutually Unbiased Bases ◽  
Identity Matrix ◽  
Polynomial Association Scheme

It was shown by LeCompte, Martin, and Owens in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite.  We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain an upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the best general upper bound for the mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.

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 Characterizations of Regularity for certain $Q$-Polynomial Association Schemes

10.37236/3260 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Sho Suda
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Mutually Unbiased Bases ◽  
Symmetric Designs

It was shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUBs) are triply regular association schemes. In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number. And we prove that a maximal set of MUBs carries a quadruply regular association scheme and characterize the quadruple regularity of MUBs by its parameter.

scholarly journals Complex Hadamard Matrices Attached to a 3-Class Nonsymmetric Association Scheme

2019 ◽  
Vol 35 (6) ◽  
pp. 1293-1304
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrices

scholarly journals A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes

10.37236/4423 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Hiroshi Nozaki ◽  
Hirotake Kurihara
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Equivalent Condition ◽  
Polynomial Property ◽  
Equivalent Conditions ◽  
Symmetric Association Scheme

We give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

scholarly journals On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Allen Herman ◽  
Gurmail Singh
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Group Rings ◽  
The 1960S ◽  
Torsion Units

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets

1999 ◽  
Vol 51 (2) ◽  
pp. 326-346 ◽  
Author(s):  
W. J. Martin ◽  
D. R. Stinson
Keyword(s):  
Linear Programming ◽  
Association Scheme ◽  
Type Theorem ◽  
Orthogonal Arrays ◽  
Association Schemes ◽  
Theoretic Approach ◽  
Linear Programming Bound ◽  
Ordered Orthogonal Arrays

AbstractIn an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

scholarly journals Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

2011 ◽  
Vol 2011 ◽  
pp. 1-29 ◽  
Author(s):  
Philip Feinsilver ◽  
John McSorley
Keyword(s):  
Linear Combination ◽  
Exponential Distribution ◽  
Trace Formula ◽  
Association Scheme ◽  
Permutation Groups ◽  
Identity Matrix ◽  
Johnson Scheme ◽  
Master Theorem

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon's master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.

scholarly journals On the equivalence between real mutually unbiased bases and a certain class of association schemes

2010 ◽  
Vol 31 (6) ◽  
pp. 1499-1512 ◽  
Author(s):  
Nicholas LeCompte ◽  
William J. Martin ◽  
William Owens
Keyword(s):  
Association Schemes ◽  
Mutually Unbiased Bases

scholarly journals On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

2020 ◽  
Vol 36 (36) ◽  
pp. 124-133
Author(s):  
Shinpei Imori ◽  
Dietrich Von Rosen
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Degrees Of Freedom ◽  
Generalized Inverse ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Identity Matrix ◽  
Wishart Matrix ◽  
The Mean ◽  
Scale Matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution.

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