In the previous work, we introduce a notion of pre-difference sets in a finite group G defined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set in \(G=NA\) with A an abelian subgroup and N a subgroup satisfying \(N\cap A=\{e\}\), from a difference set in \(N\times A\). This gives a (16, 6, 2) pre-difference set in \(D_{16}\) and a (27, 13, 6) pre-difference set in UT(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.

在先前的工作中，我们在有限组G中引入了预差分集的概念，该组由比差分集更弱的条件定义。在本文中，我们得到的预差集的结构在\（G = NA \）与甲一个交换子群和Ñ满足一个分组\（N \帽A = \ {ë\} \），从差集在\（N \ times A \）中。这在\（D_ {16} \）中设置了（16，6，2）预差，在UT中设置了（27，13，6）预差（3，3），其中不存在非平凡差异集。我们还给出了与Kesava Menon构造类似的预差分集的产品构造，该构造提供了不是差分集的无限系列的预差分集。我们显示了在具有索引2子组的组中存在预差分集的一些必要条件。对于证明，我们使用一个相当简单的框架“关系分区”，该框架是通过从关联方案中删除公理获得的。大多数结果在该框架中得到了证明。