Three-class Association Schemes Research Articles

Quasi self-dual codes over non-unital rings from three-class association schemes

<abstract><p>Let $ E $ and $ I $ denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as $ E = \langle a, b \mid 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle $ and $ I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle $. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings $ E $ and $ I $ for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of $ I $, and 4 in the case of $ E $. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over $ E $ and $ I $.</p></abstract>

Three-class association schemes from partial geometric designs

In this paper, we show that three-class association schemes can be constructed from certain partial geometric designs. First, we show that partial geometric designs with exactly three indices: r−n, λ1 and λ2 give rise to three-class association schemes, using the method of Beker and Haemers (1980). Second, we describe parameter sets of certain partial geometric designs that also give rise to three-class association schemes.

Nonsymmetric primitive translation schemes of prime power order

It is well-known that translation schemes of prime order are exactly the cyclotomic schemes. In this paper, we show that there do not exist any two-, three-, or four-class nonsymmetric primitive translation schemes of prime square order. On the other hand, we find new nonsymmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class association schemes. Moreover, we provide an affirmative answer to the following question raised by Song (J Algebr Comb 5:47---55, 1996): are there any other two-class primitive association schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for $$q \equiv 5 \hbox { (mod 8)}$$ q � 5 (mod 8) ? To be more specific, we show that a certain two-class primitive association scheme in the finite field $${\mathbb F}_{37^3}$$ F 37 3 constructed by Feng and Xiang (J Comb Theory Ser A 119:245---256, 2012) admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.

Three-class association schemes from cyclotomy

We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive.

Tetrahedral and cubical association schemes with related PBIB(3) designs

Here, two three-class association schemes called tetrahedral association scheme and cubical association scheme have been proposed along with methods of constructing partially balanced incomplete block designs based on these schemes. Designs based on cubical association scheme are found to be resolvable. An outline of the method of analysis of the designs has also been presented together with a list of PBIB(3) designs obtained using the proposed methods for number of treatments v < 100. The proposed designs are seen to be more efficient than the circular lattices (PBIB(3) designs) with the same number of experimental units for the estimation of elementary treatment effect contrasts.

Cyclotomy over products of finite fields and combinatorial applications

We introduce a new kind of cyclotomy over a cartesian product R of finitely many finite fields, which generalizes the classical cases of only one or two fields. We describe the orbits corresponding to this cyclotomy and, to a great extent, we determine the arithmetic corresponding to these orbits in the group ring Z R , i.e. given three orbits A , B and C , we study how many ways there are of expressing an element of C as a sum of two elements of A and B . In particular, we obtain the cyclotomic numbers in a variety of interesting cases. We exhibit some applications of this cyclotomy to the construction of combinatorial structures with nice groups of multipliers. More precisely, we produce an infinite family of divisible difference sets with new parameters, another family of relative difference sets, as well as some infinite families and some sporadic examples of partial difference sets. We also obtain both infinite families and a sporadic construction of three-class association schemes.

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.

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.

Geometric construction of association schemes from non-degenerate quadrics

LetF q be a finite field withq elements, whereq is a power of an odd prime. In this paper, we assume that δ=0,1 or 2 and consider a projective spacePG(2ν+δ,F q ), partitioned into an affine spaceAG(2ν+δ,F q ) of dimension 2ν+δ and a hyperplaneℋ=PG(2ν+δ−1,F q ) of dimension 2ν+δ−1 at infinity. The points of the hyperplaneℋ are next partitioned into three subsets. A pair of pointsa andb of the affine space is defined to belong to classi if the line\(\overline {ab} \) meets the subseti of ℋ. Finally, we derive a family of three-class association schemes, and compute their parameters.

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

Taking nondegenerate quadrics in projective space over finite fields of ch.=2, we construct some families of two-class and three-class association schemes, and calculate their parameters. Moreover, for the three-class association scheme, we also compute all its eigenvalues, corresponding multiplicities and the character table, and prove that it is aP-polynomial association scheme.

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.

Spreads in strongly regular graphs

A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Huffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.

Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

Taking a nondegenerate Hermitian variety as a projective set in a projective plane PG(2,s2), Mesner (1967) derived a two-class association scheme on the points of the affine space of dimension 3, for which the projective plane is the plane at infinity.We generalize his construction in two ways. We show how his construction works both for nondegenerate and degenerate Hermitian varieties in any dimension.We consider a projective space of dimension N, partitioned into an affine space of dimension N and a hyperplane H of dimension N − 1 at infinity.The points of the hyperplane are next partitioned into 2 or 3 subsets. A pair of points a,b of the affine space is defined to belong to class i if the line ab meets the subset i of H.In the first case, the two subsets of the hyperplane are a nondegenerate Hermitian variety and its complement. In this case, we show that the classification of pairs of affine points defines a family of two-class association schemes. This family of association schemes has the same set of parameters as those derived as restrictions of the Hamming association schemes to two-weight codes defined as linear spans of coordinate vectors of points on a nondegenerate Hermitian variety in a projective space of dimension N − 1. The relations of these codes to orthogonal arrays and difference sets are described in [5,6].In the second case, the three subsets are the singular point of the variety, the regular points of the variety and the complement of the variety defined by a Hermitian form of rank N − 1. This leads to a family of three-class association schemes on the points of the affine space. A geometric construction is first given for the case N = 3.Using a general algebraic method pointed out by the referee, we have also derived the three-class association scheme for general N.

Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties