Correlation Minimizing Frames

In this dissertation, we study the structure of correlation minimizing frames. A correlation minimizing (N,d)-frame is any uniform Parseval frame of N vectors in dimension, d, such that the largest absolute value of the inner products of any pair of vectors is as small as possible. We call this value the correlation constant. These frames are important as they are optimal for the 2-erasures problem. We produce the actual correlation minimizing frames. To further study the structure of correlation minimizing frames, we obtain upper bounds on the correlation constant. In the real case, we find an upper bound on the correlation constant of a correlation minimizing (N,d)-frame. As a result, we prove the correlation constant goes to zero for fixed redundancy as the dimension and number of vectors increases proportionally by 2k. When addressing the correlation constant for complex correlation minimizing (N,d)-frames, we consider circulant matrices which are also projections as the Grammian matrix of a uniform Parseval frame. We derive a relationship between these Grammian matrices and the Dirichelet kernel as well as the structure of quadratic residue. Utilizing these relationships, we obtain two upper bounds on the correlation constant. Furthermore, we investigate how the correlation constant behaves asymptotically in comparison to the Welch bound. In L[0, 1], the Laurent matrix is a projection defined by the Fourier transform of the characteristic function on an interval of fixed finite length in [0,1]. Considering the magnitude of the Fourier transform of the characteristic function on a set of su ciently small size, we derive a bound on the correlation constant and construct a method to create a correlation constant that is arbitrarily small. vi