Perfect Gaussian Integer Sequences of Odd Period <formula formulatype="inline"><tex Notation="TeX">${2^m} - 1$</tex> </formula>

A Review Study of Prime Period Perfect Gaussian Integer Sequences

“…For example, when N = 7, there exist two patterns which are the same as that of the cyclotomic class of order 2. In case of N = 31, the existing six sequence patterns are the same as that based on the Hall's sextic residue sequences [25]. There exist 18 different sequence patterns when N = 127; these patterns are different from both the Hall's sextic residue sequences and the cyclotomic class of order 2.…”

“…In addition to the fact that degree-2 PGISs can be constructed using the cyclotomic class, the same as PGISs of other degrees, many binary sequences, e.g., m-sequences and the cyclic difference set, can also be adopted to construct degree-2 PGISs, where binary sequence construction is rather a matured topic with many construction schemes or algorithms [6,25]. This implies more abundant patterns of degree-2 than other degrees.…”

“…In the case of prime period N = 4a 2 + 27 = 6 f + 1 = 2 m − 1, where a, f and m are positive integers, e.g., N = 31 and N = 127, there exist six different sequence patterns of degree-2 PGISs derived from the trace representation of Hall's sextic residue sequences [25]. (b, b, b, a, b, 0, a, a, b, 0, 0, 0, a, 0, a, 0, b, a, 0, a, 0, 0, 0, 0, a, a, 0, 0, a, 0, 0) s 11 is ternary sequence, t 2 (b, 0, 0, a, 0, a, a, 0, 0, a, a, b, a, b, 0, 0, 0, a, a, 0, a, b, b, 0, a, 0, b, 0, 0, 0, 0) t 1 to t 12 are CIDTS t 3 (b, 0, 0, b, 0, a, b, 0, 0, a, a, a, b, a, 0, 0, 0, b, a, 0, a, a, a, 0, b, 0, a, 0, 0, 0, 0) constructed based on m-sequences t 4 (b, a, a, 0, a, 0, 0, 0, a, 0, 0, a, 0, a, 0, b, a, 0, 0, 0, 0, a, a, b, 0, 0, a, b, 0, b, b) t 5 (b, 0, 0, 0, 0, b, 0, a, 0, b, b, a, 0, a, a, 0, 0, 0, b, a, b, a, a, 0, 0, a, a, 0, a, 0, 0) t 6 (b, 0, 0, a, 0, 0, a, a, 0, 0, 0, 0, a, 0, a, b, 0, a, 0, a, 0, 0, 0, b, a, a, 0, b, a, b, b) t 7 (b, b, b, 0, b, a, 0, 0, b, a, a, 0, 0, 0, 0, a, b, 0, a, 0, a, 0, 0, a, 0, 0, 0, a, 0, a, a) t 8 (b, 0, 0, 0, 0, a, 0, b, 0, a, a, a, 0, a, b, 0, 0, 0, a, b, a, a, a, 0, 0, b, a, 0, b, 0, 0) t 9 (b, a, a, 0, a, b, 0, 0, a, b, b, 0, 0, 0, 0, a, a, 0, b, 0, b, 0, 0, a, 0, 0, 0, a, 0, a, a) t 10 (b, 0, 0, b, 0, 0, b, a, 0, 0, 0, 0, b, 0, a, a, 0, b, 0, a, 0, 0, 0, a, b, a, 0, a, a, a, a) t 11 (b, a, a, a, a, 0, a, b, a, 0, 0, 0, a, 0, b, 0, a, a, 0, b, 0, 0, 0, 0, a, b, 0, 0, b, 0, 0) t 12 (b, a, a, 0, a, 0, 0, 0, a, 0, 0, b, 0, b, 0, a, a, 0, 0, 0, 0, b, b, a, 0, 0, b, a, 0, a, a) s 10 , s 8 and s 9 are constructed using cyclotomic class of orders 1, 2, and 2, respectively, and s 7 is from (12) 4.4.3.…”

“…A Gaussian integer sequence (GIS) is a sequence with coefficients that are complex numbers a + bj, where j = √ −1 and a and b are integers. The construction of a perfect Gaussian integer sequence (PGIS) has become an important research topic [21][22][23][24][25][26][27][28][29][30].…”

“…In [24], Chang et al introduced the concept of the degree of a sequence and constructed degree-2 and degree-3 PGISs of prime period p. Then, they up-sampled these PGISs by a factor of m and filled them with new coefficients to build degree-3 and degree-4 PGISs of arbitrary composite period N = mp. Lee et al focused on constructing degree-2 PGISs of various periods using two-tuple-balanced sequences and cyclic difference sets, where some PGISs are with long and high-energy efficiency properties [25][26][27][28]. Algorithms that could generate PGISs of arbitrary period were developed by Pei et al [29], and one of these algorithms could be applied to construct degree-5 PGISs of prime period p ≡ 1 (mod 4) by applying the generalized Legendre sequences (GLS).…”