Perfect sequences over the real quaternions

In this Thesis, perfect sequences over the real quaternions are first considered. Definitions for the right and left periodic autocorrelation functions are given, and right and left perfect sequences introduced. It is shown that the right (left) perfection of any sequence implies the left (right) perfection, so concepts of right and left perfect sequences over the real quaternions are equivalent. Unitary transformations of the quaternion space ℍ are then considered. Using the equivalence of the right and left perfection, it is proved that unitary transformations of the quaternion space ‘respect’ perfection of a sequence. Consequently, any symmetry transformation of the alphabet preserves perfection of a sequence. Properties of quaternionic perfect sequences are studied. It is shown that quaternionic perfect sequences share many properties in common with perfect sequences over the complex numbers. Similar to complex perfect sequences, perfection over quaternions is preserved by shifting of a perfect sequence, multiplication by a scalar, taking conjugates of each element of a perfect sequence, taking a proper decimation of a perfect sequence. However, unlike the complex case, multiplication of the elements of a perfect sequence of length n by consecutive powers of an n-root of unity destroys perfection, in general. To construct long sequences, this Thesis extends the well-known result about composition of two perfect sequences over complex numbers, of relatively prime lengths, into the domain of real quaternions. We introduce a concept of composition of two or more sequences with elements in the real quaternion algebra ℍ. Using this generalization, we construct a perfect sequence of really impressive length, in order of a few billions, over a 24-element alphabet of quaternionic 12-roots of unity. Also, a new result on composition of two sequences of even lengths is presented, and an algorithm, based on the composition two sequences of even lengths, which renders longer perfect sequences, is given. Conditions, necessary for perfection over quaternions, are studied. The Balance Theorem for the quaternions is proved, and a few generalizations of this theorem, which are also applicable to sequences over the complex numbers, are introduced. The left and the right quaternionic discrete Fourier transforms are introduced. It is shown that, dissimilar to the complex case, the property of having all discrete Fourier transform coefficients of equal norms is a necessary, but not sufficient, condition for perfection over quaternions. Many examples, illustrating new concepts and results, are given in this Thesis.