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The previous constructions of quadrature amplitude modulation (QAM) Golaycomplementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length$2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in2010 and Liu \textsl{et al.} (the generalized cases IV-V for $q\ge 3$) in 2013respectively. Those sequences are presented as the combination of thequaternary standard GCSs and compatible offsets. By providing new compatibleoffsets based on the factorization of the integer $q$, we proposed two newconstructions of $4^q $-QAM GCSs, which have the generalized cases I-V asspecial cases. The numbers of the proposed GCSs (including the generalizedcases IV-V) are equal to the product of the number of the quaternary standardGCSs and the number of the compatible offsets. For $q=q_{1}\times q_{2}\times\dots\times q_{t}$ ($q_k>1$), the number of new offsets in our firstconstruction is lower bounded by a polynomial of $m$ with degree $t$, while thenumbers of offsets in the generalized cases I-III and IV-V are a linearpolynomial of $m$ and a quadratic polynomial of $m$, respectively. Inparticular, the numbers of new offsets in our first construction is seven timesmore than that in the generalized cases IV-V for $q=4$. We also show that thenumbers of new offsets in our two constructions is lower bounded by a cubicpolynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentionedGCSs over QAM in this paper can be regarded as projections of Golaycomplementary arrays of size $2\times2\times\cdots\times2$.

New Constructions of Complementary Sequence Pairs over $4^q$-QAM