由可分组设计构造对称设计

本文提出了由一类可分组设计构造出对称设计的方法.注意到这类可分组设计的关联图对应着5类结合方案的关系图.本文利用该5类结合方案的商结合方案,由这类可分组设计构造对称设计,并举例说明了构造的具体过程.此外,提出了一种利用阵列由对称设计构造可分组设计的方法.在此基础上,证明了两个有对偶性质的可分组设计GDDDP(2,11;5;0,1)和GDDDP(2,16;6;0,1)不存在.     

利用有限仿射空间中直线构作结合方案与PBIB设计

利用有限仿射空间ＡＧ（ｎ，Ｆｑ）中的直线作为处理，在ｎ＝２和ｎ≥３时分别构作了类数为２和３的结合方案，计算了其参数，并且利用所得到的结合方案构作了若干个ＰＢＩＢ设计。     

利用伪辛空间中2维全迷向子空间构作结合方案与PBIB设计

取有限域Ｆｑ上２×２＋２维伪辛空间中２维全迷向子空间作为处理，构作了具有４个结合类和２个结合类的结合方案与ＰＢＩＢ设计，并且计算了它们的参数     

利用Z／P^αZ上矩阵构作多个结合类的结合方案与PBIB设计

本文利用有限局部环Ｚ／ｐαＺ上的１×ｎ矩阵集合构作具α个结合类的方案与ＰＢＩＢ设计，并计算了它们的参数．     

PBIB设计的约化

设Ｄ是一个有ｍ个结合类的ＰＢＩＢ设计，且它的参数λ１，λ２，．．．，λｍ中至少有两个是相等，第二节中，给出了ＰＢＩＢ设计Ｄ可以用遂次合并结合类来约化的充分必要条件，第三节给出了ＰＢＩＢ设计Ｄ可以用同时合并结合来约化的充分必要条件，最后，在第四节里我们研究了上述两种方法之间的关系。     

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．     

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．     

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

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。     

伪辛空间F_q~(2v+2+l)中一类2-维子空间的结合方案及其结构

该文利用伪辛空间Fq(2v+2+l)中一类2-维非迷向子空间构作了具有2q-1个结合类的交换 的但非对称的结合方案,并且讨论了它的结构,证明了它是其基础域上的加法群和乘法群上的 熟知的结合方案的扩张.     

伪辛空间F^(2v+2+l)_q中一类2维子空间的结合方案及其结构

该文利用伪辛空间F\-q\+\{(2v+2+l)中一类2 维非迷向子空间构作了具有2q-1个结合类的交换的但非对称的结合方案，并且讨论了它的结构，证明了它是其基础域上的加法群和乘法群上的熟知的结合方案的扩张。     

Reduction of the number of associate classes of hypercubic association schemes

The reduction of the number of associate classes of some hypercubic association schemes by clubbing certain associate classes has been studied in the paper. It has been found that the reduction of anm-class hypercubic association scheme forv=2 ^m treatments into a 2-class association scheme is always possible. Further it is proved herein that them-class hypercubic association scheme forv=s ^m treatments is reducible (i) to a 3-class association scheme, whens=3 and (ii) to a 2-class association scheme, whens=4, which really hasp ₁₁ ¹ =p ₁₁ ² and hence leads to a series of balanced incomplete block designs.     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

Association Schemes Related to Kasami Codes and Kerdock Sets

Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).     

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et al. proposed an association scheme of 64 points in R¹⁴ which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.     

Association schemes arising from bent functions

We give a construction of 3-class and 4-class association schemes from s-nonlinear and differentially 2 ^s -uniform functions, and a construction of p-class association schemes from weakly regular p-ary bent functions, where p is an odd prime.     

A Primitive Non-symmetric 3-Class Association Scheme on 36 Elements with p11 = 0 Exists and is Unique

In this paper we construct a primitive, non-symmetric 3-class association scheme with parameters v = 36, v₁ = 7, p¹₁₁ = 0 and p²₁₁ = 4 and show that such a scheme is determined by its parameters.     

A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes

A new family of distance-regular graphs is constructed. They are antipodal 2^2t–1-fold covers of the complete graph on 2^2t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs.There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.     

Association schemes related to Delsarte–Goethals codes

In this paper, we construct an infinite series of 9-class association schemes from a refinement of the partition of Delsarte–Goethals codes by their Lee weights. The explicit expressions of the dual schemes are determined through direct manipulations of complicated exponential sums. As a byproduct, another three infinite families of association schemes are also obtained as fusion schemes and quotient schemes.     

Difference set constructions of DRADs and association schemes

Doubly Regular Asymmetric Digraphs (DRAD) with rank 4 automorphism groups were previously thought to be rare. We exhibit difference sets in Galois Rings that can be used to construct an infinite family of DRADs with rank 4 automorphism groups. In addition, we construct difference sets in groups for all r?2 that can be used to construct DRADs and nonsymmetric 3-class imprimitive association schemes. Finally, we prove a new product construction for difference sets so that the resulting difference sets can be used to build nonsymmetric 3-class imprimitive association schemes.     

On a Problem of M. Kambites Regarding Abundant Semigroups

A semigroup is regular if it contains at least one idempotent in each ?-class and in each ?-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ?-class and in each ?-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ?*-class and in each ?*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ?* and ?*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ?* and ?*-class, must the idempotents commute? In this note, we provide a negative answer to this question.