Amorphic association schemes with negative Latin square type graphs

Partial difference sets from quadratic forms and $p$-ary weakly regular bent functions

We generalize the construction of affine polar graphs in two different ways to obtain new partial difference sets and amorphic association schemes. The first generalization uses a combination of quadratic forms and uniform cyclotomy. In the second generalization we replace the quadratic form in the affine polar graph construction by higher degree homogeneous functions that are $p$-ary weakly regular bent. The negative Latin square type partial difference sets arising from the first generalization are new.

每个非对称4类结合方案都可由一个有向图生成

Every nonsymmetric $4$-class association scheme can be generated by a digraph

该研究在特定假设下刻画了恰含一对非对称关系的交换结合方案的有向图生成条件，并应用该结论证明所有非对称4类结合方案都可由有向图生成，拓展了3类结合方案生成相关结论。

关于交换结合方案及关联（有向）图

On commutative association schemes and associated (directed) graphs

本文研究交换结合方案的 Bose–Mesner 代数中邻接矩阵所对应的连通有向图，给出其组合结构与代数-组合刻画，并证明任意非无定形对称的交换 3-类结合方案都可由某个有向图的邻接矩阵生成其 Bose–Mesner 代数。

Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes

For prime powers $q$ and $q+\varepsilon$ where $\varepsilon\in\{1,2\}$, an affine resolvable design from $\mathbb{F}_q$ and Latin squares from $\mathbb{F}_{q+\varepsilon}$ yield a set of symmetric designs if $\varepsilon=2$ and a set of symmetric group divisible designs if $\varepsilon=1$. We show that these designs derive commutative association schemes, and determine their eigenmatrices.

Pseudocyclic association schemes arising from the actions of PGL(2,2^m) and PΓL(2,2^m)

The action of $PGL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ affords an association scheme, which was shown to be pseudocyclic in Hollmann's thesis in 1982. It was further conjectured in Hollmann's thesis that the orbital scheme of $PΓL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ is also pseudocyclic if $m$ is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power number of points.

Association schemes for diagonal groups

For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at most $|G|$ parts, so is $p(n)$ if $|G|\ge n$; its parameters depend only on $n$ and $|G|$. For $n=2$, the association scheme is trivial, while for $n=3$ its relations are the Latin square graph associated with the Cayley table of $G$ and its complement. A transitive permutation group $G$ is said to be \emph{AS-free} if there is no non-trivial association sche…

Further Results on the Pseudo-$L_{g}(s)$ Association Scheme with $g\geq 3$, $s\geq g+2$

It is inevitable that the $L_{g}(s)$ association scheme with $g\geq 3, s\geq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $\mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s…