Non-existence and construction of pre-difference sets, and equi-distributed subsets in association schemes
Non-existence and construction of pre-difference sets, and equi-distributed subsets in association schemes
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关联方案中预差集和等分布子集的不存在和构造
DOI：
10.1007/s00373-021-02279-9
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发表时间：
2021
影响因子：
0.7
通讯作者：
Takayuki Okuda
中科院分区：
文献类型：
--
作者：
Hiroki Kajiura;Makoto Matsumoto;Takayuki Okuda
在以前的工作中，我们在有限群G中引入了预差集的概念，该有限群G由比差集更弱的条件定义。本文从中的差集出发，给出了中的一个预差集的构造，其中A是交换子群，Na是子群，满足。这给出了UT（3，3）中的（16，6，2）预差集和UT（3，3）中的（27，13，6）预差集，其中不存在非平凡差集。我们还给出了一个类似于Kesava Menon构造的预差集的乘积构造，它提供了非差集的预差集的无穷级数。给出了指数为2的子群的群中存在预差集的必要条件。为了证明，我们使用了一个相当简单的框架“关系划分”，这是通过从关联方案中删除公理而获得的。大多数结果都是在这个框架下得到验证的。
In the previous work, we introduce a notion of pre-difference sets in a finite groupGdefined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set inwithAan abelian subgroup andNa subgroup satisfying, from a difference set in. This gives a (16, 6, 2) pre-difference set inand a (27, 13, 6) pre-difference set inUT(3, 3), where no non-trivial difference sets exist. We also give a product construction of pre-difference sets similar to Kesava Menon construction, which provides infinite series of pre-difference sets that are not difference sets. We show some necessary conditions for the existence of a pre-difference set in a group with index 2 subgroup. For the proofs, we use a rather simple framework “relation partitions,” which is obtained by dropping an axiom from association schemes. Most results are proved in that frame work.