Construction of Period $qp$ PGISs With Degrees Equal to or Larger Than Four

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 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 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 ℤ_N into seven subsets and construct PGISs with more diverse degrees.