Difference Sets, Almost Difference Sets, and their Applications

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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.**