This research involves the study of some specific cases of the Correlation Problem, the instances of which are carefully chosen so that advances in any of these cases would constitute dramatic advances, and improve our understanding of the whole Correlation Problem. These cases are also chosen with an eye to practical applications. We would employ algebraic methods to generate block-Hankel weighing matrices, multi dimensional Hadamard matrices, almost difference sets and perfect sequences. Our motivation stems from their usefulness in several areas of communication engineering: quantum computing, MC-CDMA systems , quasi-synchronous CDMA , multiple antenna wireless communication systems , FHSS which are widely used in military radios, CDMA and GSM networks, radars and sonars, and Bluetooth communications - to name a few. Discrete mathematical structures that can be developed using modern algebra, number theory and finite geometry and other combinatorial structures are useful in constructing sequences and arrays with desirable correlation properties. They are systematically studied via their algebraic counterparts. The results obtained will lead to new mathematical theories that are of interest to combinatorial design theorists and communication engineers. We thus investigate sequence design problems, which have a variety of applications in communication engineering. Our methods will be very algebraic and would employ tools from algebra, finite fields, and algebraic number theory.
This effort studied some specific cases of the Correlation Problem, the instances of which were carefully chosen so that advances in any of these cases would constitute dramatic advances, and improve our understanding of the whole Correlation Problem. These cases were also chosen with an eye to practical applications. We employed algebraic methods to generate block-Hankel weighing matrices, multi dimensional Hadamard matrices, almost difference sets and perfect sequences. Our motivation stemmed from their usefulness in several areas of communication engineering: quantum computing, MC-CDMA systems , quasi-synchronous CDMA , multiple antenna wireless communication systems , FHSS which are widely used in military radios, CDMA and GSM networks, radars and sonars, and Bluetooth communications – to name a few. Intellectual merit: Discrete mathematical structures that can be developed using modern algebra, number theory and finite geometry and other combinatorial structures are useful in constructing sequences and arrays with desirable correlation properties. These mathematical ideas were successfully employed by the PI in projects supported by AFOSR, NSF and NSA, to obtain new families of perfect binary arrays and related objects. They were studied via their algebraic counterparts. The results obtained will lead to new mathematical theories that are of interest to combinatorial design theorists and communication engineers. Broader impacts: Families of sequences and arrays with good correlation properties have potential applications in cryptography and digital cellular communications. More than a dozen REU students were supported by this project. Most of them have publishable results which they presented in conferences. Some of these papers are already published and some are in the pipeline. Four women and a minority student were supported. Six different GRAs were supported, four of whom wrote thesis and have publishable results. A high school student (prodigy, on his way for PhD at University of Chicago) was supported as an REU (since he was also taking undergraduate classes) and he has one paper submitted. In a summer class, we incorporated the algebraic machinery developed here along with applications to sequence design problems. The research assistants supported major in Mathematics, Computer Engineering, Electrical Engineering, Human Factors Engineering and Computer Science (thereby improving STEM education).
