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.     

Existence of three HMOLS of type 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis><Superscript>1</Superscript>

A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by （h1n1 h2n2…hknk ）. If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS（T）. It has been proved that 3-HMOLS（2n31） exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS（2nu1） with u ≥ 4, and prove that 3-HMOLS（2~u1） exist if n ≥ 54 and n ≥7/4u ＋ 7.     

On association schemes with multiplicities 1 or a prime p

The main result of this paper gives a classification of commutative association schemes, all irreducible characters of which have multiplicity 1 or a prime p.     

Commutativity of association schemes of prime square order having non-trivial thin closed subsets

Through a study of the structure of the modular adjacency algebra over a field of positive characteristic p for a scheme of prime order p and utilizing the fact that every scheme of prime order is commutative, we show that every association scheme of prime square order having a non-trivial thin closed subset is commutative. The second author was supported by Korea Research Foundation Grant (KRF-2006-003-00008).     

CONSTRUCTING SELF-CONJUGATE SELF-ORTHOGONAL DIAGONAL LATIN SQUARES

1.IntroductionALatinsquareofordernisannxnarraysuchthateveryrowandeverycolumnisapermutationofann-setN.AtransversalinaLatinsquareisasetofpositions,oneperrowandonepercolumn,amongwhichthesymbolsoccurpreciselyonceeach.AdiagonalLatinsquareisaLatinsquarewhosemaindiagonalandbackdiagonalarebothtransversals.TwoLatinsquaresofordernareorthogonalifeachsymbolinthefirstsquaremeetseachsymbolinthesecondsquareexactlyoncewhentheyaresuperposed.ALatinsquareisself-orthogonalifitisorthogonaltoitstranspose.Inanea…     

On spherical designs obtained from Q‐polynomial association schemes

We characterize that the image of the embedding of the Q ‐polynomial association scheme into the first eigenspace by primitive idempotent E ₁ is a spherical t‐design in terms of the Krein numbers. Furthermore, we show that the strengths of P‐ and Q‐polynomial schemes as spherical designs are bounded by a constant. Copyright © 2011 John Wiley & Sons, Ltd. 19:167‐177, 2011     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

Existence of LSs

Let X be a ‐set and be a partition of X into n groups of size 2 and one group G₀ of size 4. A large‐set‐plus denoted by LS is a partition of all ‐transverse triples of X into block sets of 3‐GDD(2ⁿ4¹)s with group set , and two block sets of 3‐GDD(2ⁿ)s with group set . In this paper, we study the existence problem of LSs and give a nearly complete solution.     

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.

     

Amorphic association schemes with negative Latin square-type graphs

Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square-type graphs and whose underlying set is a nonelementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.     

Large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ have been studied by Schellenberg, Chen, Lindner and Stinson. These large sets have applications in cryptography in the construction of perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for n = 6 and for all n = 3^k, k ≥ 1. In this paper, we present new recursive constructions and use them to show that such large sets exist for all odd n ≡ 0 (mod 3) and for even n = 24m, where m odd ≥ 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 285–296, 2001     

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

Constant Weight Codes and Group Divisible Designs

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.     

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.     

Geometric construction of some families of two-class and three-class association schemes from nondegenerate quadrics in characteristic two

1.IntroductionThetheoryoflinearspacesintriteprojectivegeometryhasbeenusedbyseveralauthorsinconstructingBIBandPBIBdesigns.BoseI21firstusedthepropertiesofquadricsurfaCesinfiniteprojectivegeometryoftwoandthreedimensionsforconstr-netingexperimelltaldesigns.D.K.Ray-Chaudhurils]usedthegeometryofquadricstoconstructseveralseriesofPBIBdesignswithtwoassociateclasses.I.M.Chakravartila]usednondegenerateanddegenerateHebotianvarietiestoconstructsomefamiliesoftwo-classandthree-classassociationschemes…     

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009