Difference Sets, Almost Difference Sets, and their Applications

这个研究项目的主要目的是寻找新的差集和几乎差集，并探索它们在一些应用中的应用。差集和几乎差集是连接数论、组合学、代数和几何的有趣的组合对象。它们在工程、计算机科学和通信中有广泛的应用。当人们发现循环群上存在差集等价于存在某种具有良好自相关性质的周期序列时，人们对这一主题的兴趣大大增加。20世纪90年代，差集理论的进步开始要求使用高级代数数论。与差集相比，在过去的十年里，几乎差集的构造方面的研究要少得多。人们越来越认识到需要对几乎差集的构造进行进一步的研究。几乎差集和几乎差集在无线通信、密码学和编码理论等领域的广泛的现实世界问题中都有应用。在无线通信领域，由差集产生的序列长期以来一直应用于码分多址(CDMA)网络的信道编码理论。更多的当代研究集中在使用差集的序列来设计无线电接入协议，允许大量设备在共享的无线信道上传输，以预见未来将有数十亿无线连接的设备的物联网(IoT)范例。差集和几乎差集也应用于流密码，因为它们被用来构造高度非线性的函数，以增强安全性，例如抵抗相关攻击。它们在用于数据压缩和纠错的编码理论中也有许多应用。**

The primary objective of this research program is to seek new classes of difference sets and almost difference sets, and to explore their use in a number of applications. Difference sets and almost difference sets are interesting combinatorial objects which connect number theory, combinatorics, algebra, and geometry. They have a wide range of applications in engineering, computer science, and communications. Interest in the topic increased greatly when it was discovered that the existence of a difference set over a cyclic group is equivalent to the existence of a certain type of periodic sequence with good autocorrelation properties. ***In the 1990s, advancements in the theory of difference sets began to require the use of advanced algebraic number theory. In comparison to difference sets, significantly less research has been done on the construction of almost difference sets in the past ten years. There is growing recognition of the need to conduct further research towards the construction of almost difference sets. ***Difference sets and almost difference sets have applications to a wide range of real-world problems in areas such as wireless communications, cryptography, and coding theory. In the field of wireless communications, sequences generated from difference sets have long had applications to channel coding theory for Code-Division Multiple Access (CDMA) networks. More contemporary research focuses on using sequences from difference sets to design radio access protocols which allow a massive number of devices to transmit over a shared wireless channel, in anticipation of the Internet of Things (IoT) paradigm that foresees a future with billions of wirelessly-connected devices. Difference sets and almost difference sets also have applications to stream cipher cryptography, since they are used to construct highly non-linear functions which enhance security properties such as resistance to correlation attacks. They also have many applications to coding theory, which is used for data compression and error correction.**