论文标题
非亚伯2组中差异集的结构
Constructions of difference sets in nonabelian 2-groups
论文作者
论文摘要
差异集已研究了80多年。代数数理论，群体理论，有限几何和数字通信工程的技术已被用来建立建设性和不存在的结果。我们提供了一种新的理论方法，该方法通过完善戴维斯（Davis）和杰迪瓦布（Jedwab）在1997年引入的扩展建筑集的概念，从而大幅度扩展了$ 2 $ - 组的$ 2 $ - 组类别。然后，我们将如何使用产品结构和其他方法来描述在剩余的2美元$ 2 $ groups中构建差异集。我们宣布完成了十年的合作工作，以精确确定56,092个非同态组中的哪个订单256包含差异集。所有两个经典不存在标准不排除的命令256组都包含一个差异集，与先前的第4、16和64组组的调查结果一致。我们为如何解决所有订单$ 2 $ - 组中差额集的存在问题的建议。
Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of $2$-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining $2$-groups. We announce the completion of ten years of collaborative work to determine precisely which of the 56,092 nonisomorphic groups of order 256 contain a difference set. All groups of order 256 not excluded by the two classical nonexistence criteria are found to contain a difference set, in agreement with previous findings for groups of order 4, 16, and 64. We provide suggestions for how the existence question for difference sets in $2$-groups of all orders might be resolved.
