Sung Y. Song

Polygonal designs: some existence and non-existence results

Abstract Polygonal designs are introduced as a generalization of balanced incomplete block designs and as a specialization of partially balanced incomplete block designs. As in the case of balanced incomplete block designs, there is no hope of deciding the values of the parameters for which polygonal designs exist. We develop enough theory to reveal the structure, and thus, to resolve the existence problem for small polygonal designs, and derive necessary conditions for general cases.

Association schemes of small order

Abstract We survey the work on the classification problem of association schemes of small order. Our aim is to introduce some construction methods which simplify the description of known schemes, and provide the complete catalogue of all association schemes for a given order n, where n=4,5,…,15. Part of this work deals with the construction and enumeration of association schemes via Schur rings and two ways of tensoring association matrices; another part deals with the fusion relation of association schemes of a given order by using the notion of the wreath product and direct product of association schemes. The association schemes are presented in Hasse diagrams of partially ordered sets under the fusion relations. Our list includes, in particular, all symmetric association schemes, the source of partially balanced incomplete block designs with the corresponding number of treatments.

On a class of partially balanced incomplete block designs with applications in survey sampling

We revisit the existence and construction problems of balanced sampling plans excluding contiguous units, a class of designs introduced and studied by Hedayat, Rao, and Stufken (1988a,b). A link between these designs and a class of partially balanced incomplete block designs is established, and a Fisher-type inequality is shown to hold. Special attention is given to those designs that attain equality in this Fisher-type inequality. Some new designs are obtained by construction, while the nonexistence of designs is considered for some other sets of parameters.

Character Tables of Fission Schemes and Fusion Schemes

For given commutative association schemes H = (X, {Ri}0?i?d and Y = (X, {Sj}0?j?e, if {Ri}0?i?d is a refinement of {Sj}0?j?e, then we say that X is a fission scheme of Y. It is shown that if each class Sj of Y is either equal to Ri for some i or obtained by fusing two Ris in a simple uniform pattern, then the character table of one is determined by the other. This fact is proved by orthogonality relation of the character tables. Then the character table of the group scheme coming from the action of Chevalley group G2 (q) on the set ?± of hyperplanes of type O±6(q) in the seven-dimensional orthogonal geometry over GF(q) is calculated from the character table of its fusion scheme, which is obtained from the action of O7(q) on ?±, respectively. Further application of the main result is discussed in various other examples.

Fusion Relation in Products of Association Schemes

Abstract. Fusion relations between the association schemes obtained by direct product and wreath product are established via a study of their matrix representations. The character table of the scheme obtained by the wreath product is described and some algebraic properties of the products are derived.

Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures

We present many new directed strongly regular graphs by direct construction. We construct these graphs on the collections of antiflags of certain finite incidence structures. In this way, we confirm the feasibility of infinitely many parameter sets that was previously undetermined. We describe some examples of graphs together with their isomorphism classes to demonstrate the fact that our construction method is capable of producing many graphs with same parameters.

Relative efficiencies of sampling plans for selecting a small number of units from a rectangular region

We examine the efficiency of several sampling plans for use in certain agricultural, ecological and environmental studies. One concern for such studies is that plots that arephysically close might be more similar than distant plots. We considered sampling plansthat are designed to generate samples that represent the entire population while avoidingthe selection of units that provide essentially redundant information. All plans havethe property that they avoid the simultaneous selection of units that are, in some sense,neighboring units. By means of a simulation study, the efficiency of these plans iscompared to simple random Aampling Factors that influence the relative efficiencies areexamined. This is done for a number of different populations, representing variouspossible patterns for a response variable.

Class 3 association schemes whose symmetrizations have two classes

Abstract For a given commutative association scheme L , by fusing all the non-self-paired relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme L. In this paper, we investigate when a symmetric association scheme L of class 2 admits a symmetrizable commutative fission scheme L of class 3. By studying the feasible parameter sets obtained from the acceptible fission character tables of given symmetric schemes, we discover that there are infinitely many symmetric association schemes of class 2, each of which admits one or two symmetrizable commutative fission schemes of class 3. We then characterize and classify some such symmetric schemes and their symmetrizable commutative fission schemes. Also we prove that none of the Hamming schemes H(2, q) with q ≥ 3, and none of the Johnson schemes J(ν, 2) with ν ≥ 5 and ν ≢ 7(mod 8) have such commutative fission schemes of class 3. H(2, 2) has a symmetrizable commutative fission scheme which is isomorphic to the group scheme L ( Z 4), and J(4, 2) has one such fission scheme coming from the action of the alternating group A 4 on the set of two-element subsets from a four-element set.

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.

Characters of finite quasigroups VI: critical examples and doubletons

Examples of association schemes coming from symmetric group actions on doubletons are shown to have character tables which are not character tables of quasigroups. The examples are critical in separating the character theory of quasigroups from the general theory of association schemes.