New Algebraic Techniques for Constructing Sequences and Arrays with Good Correlation Properties

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New Algebraic Techniques for Constructing Sequences and Arrays with Good Correlation Properties K.T.ArasuDepartment of Mathematics & Statistics, Wright State University,Dayton, OH 45435 The Correlation Problem covers a broad fundamental combinatorial problem with deep mathematical content. Specific solutions to the Correlation Problem have practical diverse applications in communications, experimental design, laboratory instrumentation and manufacturing. As technology changes so too will the instances of the Correlation Problem which need solving. This research develops new mathematical techniques for attacking the problem. The Correlation Problem is to design sequences or arrays with specified dimensions with entries chosen from a specified finite set so that all non-trivial periodic autocorrelations lie in a prescribed restrictive set. Usually the autocorrelations are computed using a quadratic form. More generally, one may seek several arrays with good autocorrelations and good cross-correlations. The Correlation Problem divides into two questions: When do solutions exist, and if so, what does the solution space look like? The sequence design problems that we investigate have a variety of applications in communication engineering. The methods used will be very algebraic and would employ tools from algebra, finite fields, algebraic number theory and representation theory. Calculation of the linear span (p-ranks) of the obtained sequences will use combinatorial tools, in conjunction with the algebraic tools developed here.