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Quadrature amplitude modulation (QAM) constellation and Golay complementary sequences (GCSs) are usually applied in orthogonal frequency division multiplexing (OFDM) systems to obtain a higher data rate and a lower peak-to-mean envelope power ratio (PMEPR). In this paper, after a sufficient search of the literature, it was found that increasing the family size is an effective way to improve the data rate, and the family size is mainly determined by the number of offsets in the general structure of QAM GCSs. Under the guidance of this idea, we propose a new construction for $4^q$-QAM GCSs through generalized Boolean functions (GBFs) based on a new description of a $4^q$-QAM constellation, which aims to enlarge the family size of GCSs and obtain a low PMEPR. Furthermore, a previous construction of $4^q$-QAM GCSs presented by Li has been proved to be a special case of the new one, and the family size of new sequences is much larger than those previously mentioned, which means that there was a great improvement in the data rate. On the other hand, a previous construction of 16-QAM GCSs presented by Zeng is also a special case of the new one in this paper, when $q=2$. In the meantime, the proposed sequences have the same PMEPR upper bound as the previously mentioned sequences presented by Li when applied in OFDM systems, which increase the data rate without degrading the PMEPR performance. The theoretical analysis and simulation results show that the proposed new sequences can achieve a higher data rate and a low PMEPR. Full article