Construction of Period <inline-formula> <tex-math notation="LaTeX">$qp$ </tex-math> </inline-formula> PGISs With Degrees Equal to or Larger Than Four

Abstract

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of $N = qp$ , where $q$ and $p$ are distinct primes. The study begins with the partitioning of a ring $\mathbb {Z}_{N}$ into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of $\mathbb {Z}_{N}$ should be carried out; here, two cases, the even period $N = 2p$ and the odd period $N = qp$ , are treated separately. We can adopt the Legendre sequences of the prime period $p$ to construct PGISs of period $2p$ with degrees larger than four. For the case of period $qp$ , we introduce the Jacobi symbols to partition $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.