Perfect sequences over the real quaternions

Abstract

In this paper, perfect sequences over real quaternions are 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.