Schematic 4-designs

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.     

Three-Class Association Schemes

We study (symmetric) three-class association schemes. The graphs with four distinct eigenvalues which are one of the relations of such a scheme are characterized. We also give an overview of most known constructions, and obtain necessary conditions for existence. A list of feasible parameter sets on at most 100 vertices is generated.     

Three-Class Association Schemes on Galois Rings in Characteristic 4

Small feasible parameters for ``nontrivial' symmetric three-class association schemes were listed in a paper of van Dam. This work addresses the question of existence for three-class association schemes on 4ⁿ vertices. Several families of three-class symmetric schemes are constructed in association schemes of Galois rings of characteristic 4.     

特征为2的奇异二次面上3个类和4个类结合方案的几何构作(英文)

利用特征为２的有限域上射影空间,我们构作了一些三个类和４个类的结合方案,并计算了它们的参数．     

Classification of three-class association schemes using backtracking with dynamic variable ordering

We describe several techniques for the exhaustive computer generation of non-isomorphic association schemes with a given set of intersection numbers using a backtracking algorithm with forward checking and dynamic variable ordering. We have applied these techniques to the classification of certain open parameter sets for three-class association schemes listed by Van Dam in (Three-class association schemes, J. Algebraic Combin. 10 (1999) 69–107) for which we present several new results. Among these are some new (imprimitive) distance regular graphs of diameter 3.     

Three-valued Gauss periods,circulant weighing matrices and association schemes

Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt and White and others. In this paper, we consider the question of when Gauss periods take exactly three rational values. We obtain numerical necessary conditions for Gauss periods to take exactly three rational values. We show that in certain cases, the necessary conditions obtained are also sufficient. We give numerous examples where the Gauss periods take exactly three values. Furthermore, we discuss connections between three-valued Gauss periods and combinatorial structures such as circulant weighing matrices and three-class association schemes.     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

Spreads in Strongly Regular Graphs

A spread of a strongly regular graph is a partitionof the vertex set into cliques that meet Delsarte's bound (alsocalled Hoffman's bound). Such spreads give rise to coloringsmeeting Hoffman's lower bound for the chromatic number and tocertain imprimitive three-class association schemes. These correspondenceslead to conditions for existence. Most examples come from spreadsand fans in (partial) geometries. We give other examples, includinga spread in the McLaughlin graph. For strongly regular graphsrelated to regular two-graphs, spreads give lower bounds forthe number of non-isomorphic strongly regular graphs in the switchingclass of the regular two-graph.     

On non-symmetric commutative association schemes with exactly one pair of non-symmetric relations

For a given non-symmetric commutative association scheme, by fusing all the non-symmetric relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme. In this paper, we introduce a set of feasibility and realizability conditions for a class e symmetric association scheme to be split into a class e+1 non-symmetric commutative association scheme. By applying the feasibility and realizability conditions, we obtain a classification into six categories of the class 4 non-symmetric fission schemes of group-divisible 3-schemes. Complete solutions for three of the six categories and partial results for the remaining cases are presented.     

Character Tables of Association Schemes Based on Attenuated Spaces

The set of subspaces of a given dimension in an attenuated space has a structure of a symmetric association scheme and this association scheme is called an association scheme based on an attenuated space. Association schemes based on attenuated spaces are generalizations of Grassmann schemes and bilinear forms schemes, and also q-analogues of nonbinary Johnson schemes. Wang, Guo, and Li computed the intersection numbers of association schemes based on attenuated spaces. The aim of this paper is to compute character tables of association schemes based on attenuated spaces using the method of Tarnanen, Aaltonen, and Goethals. Moreover, we also prove that association schemes based on attenuated spaces include as a special case the m-flat association scheme, which is defined on the set of cosets of subspaces of a constant dimension in a vector space over a finite field.     

Characters of association schemes containing a strongly normal closed subset of prime index

We consider a relation between characters of an association scheme and its strongly normal closed subsets with prime index. As an application of our result, we show that an association scheme of prime square order with a proper strongly normal closed subset is commutative.

     

Strongly Regular Decompositions of the Complete Graph

We study several questions about amorphic association schemes and other strongly regular decompositions of the complete graph. We investigate how two commuting edge-disjoint strongly regular graphs interact. We show that any decomposition of the complete graph into three strongly regular graphs must be an amorphic association scheme. Likewise we show that any decomposition of the complete graph into strongly regular graphs of (negative) Latin square type is an amorphic association scheme. We study strongly regular decompositions of the complete graph consisting of four graphs, and find a primitive counterexample to A.V. Ivanov's conjecture which states that any association scheme consisting of strongly regular graphs only must be amorphic.     

New parameters of subsets in polynomial association schemes

We define new parameters, a zero interval and a dual zero interval, of subsets in P- or Q-polynomial association schemes. A zero interval of a subset in a P-polynomial association scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a Q-polynomial association scheme is a successive interval index for which the dual inner distribution vanishes. We derive bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a P-polynomial association scheme (resp. a Q-polynomial association scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a Q-polynomial association scheme). Moreover, we consider the spherical analogue of a dual zero interval.     

AN EXAMPLE OF THE NON-UNIQUENESS OF THE L_5(10) ASSOCIATION SCHEME

It is known from Bose and Shimamoto that the existence of a Latin square typeL_5(10)association scheme with parameters v=100,n_1=45,p_(11)~1=20,and p_(11)~2=20 presupposesthe existence of a set of three mutually orthogonal Latin squares of order 10.But as yet wehave not known whether such a set of orthogonal Latin squares exists.In this note the author gives an association scheme with the above parameters which isnot an L_5(10)association scheme.     

Algebraic structure of association schemes of prime order

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.     

On higher indicators of regular association schemes

In the present paper, we will define the higher Frobenius–Schur indicators and the higher indicators of association schemes as a generalization of those of finite groups. The higher indicators of any association scheme are always positive rational numbers. Especially, for any positive integer $n$, the $n$th indicator of any regular association scheme is the number of relations such that its strong girth divides $n$. Thus, all higher indicators of any regular association scheme are natural numbers, and the sequence of the indicators is periodic. We will show that the converses of these facts are also true for finite exponent association schemes. Finally, we introduce a family of infinite exponent association schemes all higher indicators of which are natural numbers and the sequence of the indicators of which is periodic.     

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.     

An analogue oft-Designs in the association schemes of alternating bilinear forms

The notion of designs in an association scheme is defined algebraically by Delsarte [4]. It is known that his definition of designs has a geometric interpretation for known (P andQ)-polynomial association schemes except three examples. In this paper we give a geometric interpretation of designs in an association scheme of alternating bilinear forms, which is one of the three.     

特征为2的奇异二次面上3个类和4个类 -4 结合方案的几何构作

利用特征为 2 的有限域上射影空间,我们构作了一些三个类和 4 个类的结合方案,并计算了它们的参数．     

Association schemes based on attenuated spaces

The subspaces of a given dimension in an attenuated space form the points of an association scheme. If the dimension is maximal, it is the bilinear forms graph, which has been thoroughly studied. In this paper, we discuss the general case, and obtain a family of symmetric association schemes which are a common generalization of the Grassmann schemes and the bilinear forms schemes. Moreover, all the parameters of the association scheme are computed.