  In this paper, we firstly propose a novel construction of $16$-quadrature
amplitude modulation (QAM) near-complementary sequences with low peak-to-mean
envelope power ratio (PMEPR) in orthogonal frequency division multiplexing
(OFDM) systems. The proposed $16$-QAM near-complementary sequences can be
constructed by utilizing novel nonlinear offsets, where the length of the
sequences is $n=2^m$. The family size of the newly constructed $16$-QAM
near-complementary sequences is $8\times (\frac{m!}{2})\times 4^{m+1}$, and the
PMEPR of these sequences is proven to satisfy ${\textrm{PMEPR}}\leq 2.4$. Thus,
the proposed construction can generate a number of $16$-QAM near-complementary
sequences with low PMEPR, resulting in the improvement of the code rate in OFDM
systems. Furthermore, we also propose a novel construction of $64$-QAM
near-complementary sequences with low PMEPR, which is the first proven
construction of $64$-QAM near-complementary sequences. The PMEPRs of two types
of the proposed $64$-QAM near-complementary sequences are proven to satisfy
that ${\textrm{PMEPR}}\leq 3.62$ or ${\textrm{PMEPR}}\leq 2.48$, respectively.
The family size of the newly constructed $64$-QAM near-complementary sequences
is $64\times (\frac{m!}{2})\times 4^{m+1}$.
