This paper presents a method for constructing degree-(k + 1) perfect Gaussian integer sequence (PGIS) of period N = pk, where p is a prime number. The study begins with the partitioning of set Z n into k + 1 subsets and exploration of their properties and theorems. The base sequences can be defined and the associated k + 1 degree PGIS is constructed based on the partitioning of Z N . The k constraint equations that govern the k + 1 different sequence coefficients to match the criteria for a sequence to be perfect are nonlinear equations, which makes the construction of higher degree PGISs especially challenging. A new method of transforming k nonlinear constraint equations into 2 k − 2 linear equations with 2 k − 2 variables is presented. It is then easy to derive a unique solution, from which the construction of degree-(k + 1) PGISs becomes straightforward. Both degree-5 and degree-7 PGIS examples are provided for demonstration.