Perfect Gaussian Integer Sequences of Arbitrary Composite Length

Abstract

A composite number can be factored into either N=mp or N=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where p is an odd prime and m, n ≥ 2 are integers. This paper proposes a method for constructing degree-3 and degree-4 perfect Gaussian integer sequences (PGISs) of an arbitrary composite length utilizing an upsampling technique and the base sequence concept proposed by Hu, Wang, and Li. In constructing the PGISs, the degree of the sequence is defined as the number of distinct nonzero elements within one period of the sequence. This paper commences by constructing degree-3 PGISs of odd prime length, followed by degree-2 PGISs of odd prime length. The proposed method is then extended to the construction of degree-3 and degree-4 PGISs of composite length N=mp. Finally, degree-3 and degree-4 PGISs of length N=4 are built to facilitate the construction of degree-3 and degree-4 PGISs of length N=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where n ≥ 3.