Perfect Gaussian Integer Sequences of Period $p^{k}$ With Degrees Equal to or Less Than $k+1$

This paper presents the construction of perfect Gaussian integer sequence (PGIS) of period N = p
^k
with degrees equal to or less than k + 1, where p is a prime number and k ≥ 2. The study begins with the partitioning of ℤ
_N
into k + 1 subsets, from which k+1 base sequences are defined to construct degree-(k + 1) PGISs. The k constraint equations derived from matching the ideal periodic autocorrelation function criterion in the time domain are nonlinear. We propose a decomposition method that can transform these k nonlinear equations into 2k-2 linear equations with 2k -2 variables. The number of nonlinear constraint equations derived from the frequency domain is k +1, where the k +1 sequence coefficients can be derived sequentially by an iteration algorithm. These two proposed schemes simplify the construction of the degree-(k + 1) PGIS. PGISs having less than k + 1 degrees can be constructed in two steps. First, some sequence coefficients are assigned a zero value or one or more pairs of two consecutive base sequences are combined into one. The second step involves the adjustment of new sequence coefficients for the associated sequence to be perfect. We show that there exist at least 2k + ((k(k - 1)(k - 2))/6) distinct PGIS patterns of period N = p
^k
with degrees equal to or less than k+1.