The researches of Golay complementary sequences (GCSs) over 16 and 64 quadrature amplitude modulation (QAM) from 2001 to 2008 were generalized to$4^{q} $-QAM GCSs of length$2^{m}$by Li (the generalized cases I-III for$q\ge 2$) in 2010 and Liu et al. (the generalized cases IV-V for$q\ge 3$) in 2013. Those sequences are presented by the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer$q$, we propose two new constructions of$4^{q} $-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. Denote the number of prime factors of$q$counted with multiplicity by$\Omega (q)$. The number of new offsets in our first construction is lower bounded by a polynomial of$m$with degree$\Omega (q)$, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial and a quadratic polynomial of$m$, respectively. If$q$has a prime factor larger than 2, the number of new offsets in our second construction is lower bounded by a polynomial of$m$with degree$\Omega (q)+1$. As an example, the new offsets in our two constructions for$q=6$, whose number is bounded by a cubic polynomial, is also given. The proof in this paper implies that all the mentioned GCSs over QAM can be regarded as projections of Golay complementary arrays of size$2\times 2\times \cdots \times 2$.

A new method to construct q-ary complementary sequence sets (CSSs) and complete complementary codes (CCCs) of size N is proposed by using desired para-unitary (PU) matrices.The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given.A general form of these functions only depends on a basis of the functions from ZN to Zq and representatives in the equivalent class of Butson-type Hadamard (BH) matrices.Especially, the realization of Golay pairs from the our general form exactly coincides with the standard Golay pairs.The realization of ternary complementary sequences of size 3 is first reported here.For the realization of the quaternary complementary sequences of size 4, almost all the sequences derived here are never reported before.Generalized seed PU matrices and the recursive constructions of the desired PU matrices are also studied, and a large number of new constructions of CSSs and CCCs are given accordingly.From the perspective of this paper, all the known results of CSSs and CCCs with explicit GBF form in the literature (except non-standard Golay pairs) are constructed from the Walsh matrices of order 2.This suggests that the proposed method with the BH matrices of higher orders will yield a large number of new CSSs and CCCs with the exponentially increasing number of the sequences of low peak-to-mean envelope power ratio.

The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as 4q-QAM GCSs of length 2mby Li (the generalized cases I-III for q ≥ 2) in 2010 and Liu (the generalized cases IV-V for q ≥ 3) in 2013 respectively. Those sequences are given by the weighted sum of q quaternary standard GCSs, which is represented as q-dimensional vectorial generalized Boolean functions (V-GBFs). In this paper, we present a new construction for 4q-QAM GCSs of length 2m. The new construction includes the generalized cases I-III as special cases. If q is a composite number, a great number of new GCSs other than the sequences in the generalized cases I-V will arise. For the cases q = 4 and q = 6, we show that the ratios of the number of new GCSs and the generalized cases I-V are greater than seven and six respectively if m is large enough.