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 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 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 Two-valenced association schemes and the Desargues theorem

2019 ◽  
Vol 9 (3) ◽  
pp. 481-493
Author(s):  
Mitsugu Hirasaka ◽  
Kijung Kim ◽  
Ilia Ponomarenko
Keyword(s):  
Noncommutative Geometry ◽  
Association Scheme ◽  
Technical Condition ◽  
Association Schemes ◽  
Sufficient Condition ◽  
Desargues Theorem

AbstractThe main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any $$\{1,p\}$$ { 1 , p } -scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

Class dimension of association schemes in singular linear spaces

2017 ◽  
Vol 09 (01) ◽  
pp. 1750005
Author(s):  
Haixia Guo ◽  
You Gao
Keyword(s):  
Association Scheme ◽  
Upper Bounds ◽  
Association Schemes ◽  
Resolving Set ◽  
Linear Spaces ◽  
Set Of Points ◽  
Special Case ◽  
Singular Linear Spaces

A resolving set for an association scheme [Formula: see text] is a set of points [Formula: see text] such that, for all [Formula: see text], the ordered list of relations [Formula: see text] uniquely determines [Formula: see text], where [Formula: see text] denotes the relation [Formula: see text] containing the pair [Formula: see text] in [Formula: see text]. In this paper, we determine upper bounds on class dimension for a family of association schemes in singular linear spaces, and construct their resolving sets for a special case.

scholarly journals On the Connectivity of Graphs in Association Schemes

10.37236/6820 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Brian G. Kodalen ◽  
William J. Martin
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Association Scheme ◽  
Undirected Graph ◽  
Association Schemes ◽  
Vertex Connectivity ◽  
Edge Connectivity

Let $(X,\mathcal{R})$ be a commutative association scheme and let $\Gamma=(X,R\cup R^\top)$ be a connected undirected  graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $\Gamma$ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two vertices $a,b\in X$  are called "twins" in $\Gamma$ if they have identical neighborhoods: $\Gamma(a)=\Gamma(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in $\Gamma$ results in a connected graph. Using this and other tools, we find lower bounds on the connectivity of $\Gamma$, especially in the case where $\Gamma$ has diameter two.

On the association schemes with the thin radical series

2019 ◽  
Vol 18 (04) ◽  
pp. 1950071
Author(s):  
Javad Bagherian
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Wedge Product

In this paper, we first show that the wedge product of a thin association scheme and a schurian association scheme is schurian. Then as an application of this result, we investigate the schurity problem for the association schemes having the thin radical series. We show that these association schemes are schurian under some conditions on the successive quotients of their thin radical series.

scholarly journals On Relative $t$-Designs in Polynomial Association Schemes

10.37236/4889 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Eiichi Bannai ◽  
Etsuko Bannai ◽  
Sho Suda ◽  
Hajime Tanaka
Keyword(s):  
Lower Bounds ◽  
Association Scheme ◽  
Point Of View ◽  
Association Schemes ◽  
Algebraic Characterization ◽  
Terwilliger Algebra ◽  
C Algebra ◽  
Irreducible Modules

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.

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 $4$.

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