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The researches of Golay complementary sequences (GCSs) over 16 and 64 quadrature amplitude modulation (QAM) from 2001 to 2008 were generalized to <inline-formula> <tex-math notation="LaTeX">$4^{q} $ </tex-math></inline-formula>-QAM GCSs of length <inline-formula> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> by Li (the generalized cases I-III for <inline-formula> <tex-math notation="LaTeX">$q\ge 2$ </tex-math></inline-formula>) in 2010 and Liu et al. (the generalized cases IV-V for <inline-formula> <tex-math notation="LaTeX">$q\ge 3$ </tex-math></inline-formula>) 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 <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>, we propose two new constructions of <inline-formula> <tex-math notation="LaTeX">$4^{q} $ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> counted with multiplicity by <inline-formula> <tex-math notation="LaTeX">$\Omega (q)$ </tex-math></inline-formula>. The number of new offsets in our first construction is lower bounded by a polynomial of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> with degree <inline-formula> <tex-math notation="LaTeX">$\Omega (q)$ </tex-math></inline-formula>, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial and a quadratic polynomial of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, respectively. If <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> has a prime factor larger than 2, the number of new offsets in our second construction is lower bounded by a polynomial of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> with degree <inline-formula> <tex-math notation="LaTeX">$\Omega (q)+1$ </tex-math></inline-formula>. As an example, the new offsets in our two constructions for <inline-formula> <tex-math notation="LaTeX">$q=6$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$2\times 2\times \cdots \times 2$ </tex-math></inline-formula>.

New Constructions of Complementary Sequence Pairs Over 4q-QAM

New Constructions of Complementary Sequence Pairs Over 4^q-QAM