Perfect complementary sequence sets (PCSSs) are widely used in multi-carrier code-division multiple-access (MC-CDMA) communication system. However, the set size of a PCSS is upper bounded by the number of row sequences of each two-dimensional matrix in PCSS. Then quasi-complementary sequence set (QCSS) was proposed to support more users in MC-CDMA communications. For practical applications, it is desirable to construct an $(M,K,N,\vartheta_{max})$-QCSS with $M$ as large as possible and $\vartheta_{max}$ as small as possible, where $M$ is the number of matrices with $K$ rows and $N$ columns in the set and $\vartheta_{max}$ denotes its periodic tolerance. There exists a tradoff among these parameters and constructing QCSSs achieving or nearly achieving the known correlation lower bound has been an interesting research topic. Up to now, only a few constructions of asymptotically optimal or near-optimal periodic QCSSs were reported in the literature. In this paper, we construct five families of asymptotically optimal or near-optimal periodic QCSSs with large set sizes and low periodic tolerances. These families of QCSSs have set size $\Theta(q^2)$ or $\Theta(q^3)$ and flock size $\Theta(q)$, where $q$ is a power of a prime. To the best of our knowledge, only three known families of periodic QCSSs with set size $\Theta(q^2)$ and flock size $\Theta(q)$ were constructed and all other known periodic QCSSs have set sizes much smaller than $\Theta(q^2)$. Our new constructed periodic QCSSs with set size $\Theta(q^2)$ and flock size $\Theta(q)$ have better parameters than known ones. They have larger set sizes or lower periodic tolerances.The periodic QCSSs with set size $\Theta(q^3)$ and flock size $\Theta(q)$ constructed in this paper have the largest set size among all known families of asymptotically optimal or near-optimal periodic QCSSs.

翻译：完美互补序列集在多载波码分多址通信系统中被广泛应用。然而，完美互补序列集的集合规模受限于其中每个二维矩阵的行序列数量。为此，拟互补序列集被提出以支持更多用户接入多载波码分多址通信系统。在实际应用中，需要构造具有尽可能大的$M$值和尽可能小的$\vartheta_{max}$值的$(M,K,N,\vartheta_{max})$-拟互补序列集，其中$M$表示集合中具有$K$行$N$列的矩阵数量，$\vartheta_{max}$表示其周期互相关容限。这些参数之间存在权衡关系，构造达到或接近已知相关下界的拟互补序列集一直是一个重要的研究课题。迄今为止，文献中仅报道了少数渐近最优或近最优周期拟互补序列集的构造方法。本文构造了五个具有大规模集合尺寸和低周期互相关容限的渐近最优或近最优周期拟互补序列集族。这些序列集族具有$\Theta(q^2)$或$\Theta(q^3)$的集合规模以及$\Theta(q)$的群组规模，其中$q$为素数的幂。据我们所知，目前仅构造出三个已知的具有$\Theta(q^2)$集合规模和$\Theta(q)$群组规模的周期拟互补序列集族，而其他已知周期拟互补序列集的集合规模均远小于$\Theta(q^2)$。本文新构造的具有$\Theta(q^2)$集合规模和$\Theta(q)$群组规模的周期拟互补序列集在参数上优于已知序列集，它们具有更大的集合规模或更低的周期互相关容限。本文构造的具有$\Theta(q^3)$集合规模和$\Theta(q)$群组规模的周期拟互补序列集，在所有已知的渐近最优或近最优周期拟互补序列集族中具有最大的集合规模。