Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic $(p^{2n}, p^n-1, p^n-1, p^n+1)$ QCSSs with small alphabet size $p$, which has larger set size than the known family of periodic $(p^n(p^n-1), p^n-1, p^n-1, p^n+1)$ QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size $\Theta(K^2)$ for some flock size $K$. Compared with known asymptotically optimal aperiodic QCSSs in the literature, the proposed aperiodic QCSSs by us have better parameters or new lengths of their constituent sequences.

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