The fact that the groups \({\mathbb {Z}}_{2^m} \times {\mathbb {Z}}_{2^m}\) contain difference sets was first established by induction by Jim Davis in his Virginia dissertation of 1987. Later that year we gave a direct construction for a very large family of highly structured inequivalent difference sets in these groups. In this paper, we give a proof of our result which we presented long ago in a colloquium at Wright State University, but which has never been published. While the proof lays bare the rich structure of the difference sets, its utter simplicity suggests that it might just be from the book. We also provide some historical background and elaborate on some of the structural properties of these difference sets which have motivated some recent progress toward the goal of classifying all hadamard 2-groups.

组\（{\ mathbb {Z}} _ {2 ^ m} \ times {\ mathbb {Z}} _ {2 ^ m} \）的事实包含差异集最初是由吉姆·戴维斯（Jim Davis）在1987年的弗吉尼亚论文中通过归纳法建立的。那年晚些时候，我们对这些组中的非常大的一组高度结构化的不等价差异集进行了直接构造。在本文中，我们证明了我们的结果，该结果是我们很久以前在赖特州立大学的一次座谈会上提出的，但从未发表过。虽然证明不完整，但差异非常大，表明它可能只是本书中的内容。我们还提供了一些历史背景，并详细说明了这些差异集的某些结构性质，这些动机促使人们朝着将所有hadamard 2组进行分类的目标取得了一些新进展。