Difference Sets, Almost Difference Sets, and their Applications

差集、近似差集及其应用

• 批准号: RGPIN-2015-05208
• 负责人: Alaca, Saban
• 金额: $ 0.8万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613414
• 关键词: Difference; Sets; Almost; their; Applications

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.

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。