Difference Sets, Almost Difference Sets, and their Applications

本研究计划的主要目标是寻找新的差分集和几乎差分集的类别，并探索它们在许多应用中的用途。差分集和近似差分集是连接数论、组合学、代数和几何的有趣组合对象。它们在工程、计算机科学和通信领域有着广泛的应用。当发现循环群上的一个差集的存在性等价于具有良好自相关性质的某一类周期序列的存在性时，人们对这个问题的兴趣大大增加了。***在20世纪90年代，差分集理论的进展开始要求使用高级代数数论。与差分集相比，近十年来对几乎差分集构造的研究明显较少。越来越多的人认识到有必要对几乎差集的构造进行进一步的研究。***差分集和几乎差分集在诸如无线通信、密码学和编码理论等领域广泛应用于现实世界的问题。在无线通信领域，由差分集产生的序列在码分多址（CDMA）网络的信道编码理论中早就有了应用。更现代的研究侧重于使用来自不同集合的序列来设计无线电接入协议，该协议允许大量设备通过共享无线信道传输，以预测物联网（IoT）范式，该范式预见了数十亿无线连接设备的未来。差分集和几乎差分集在流密码密码学中也有应用，因为它们用于构造高度非线性的函数，从而增强了安全性，例如抵抗相关攻击。它们在编码理论中也有许多应用，用于数据压缩和纠错

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.**