Perfect complementary sequence sets (PCSSs) are widely used in multi-carrier code-division multiple-access (MC-CDMA) communication systems. However, the set size of a PCSS is upper bounded by the number of row sequences of each two-dimensional matrix in the PCSS. Then quasi-complementary sequence sets (QCSSs) were 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 tradeoff among these parameters. 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 have been reported in the literature. In this paper, based on polynomials over finite fields and Gaussian sums, we construct five new families of asymptotically optimal or near-optimal periodic QCSSs with large set sizes and low periodic tolerances. These families of QCSSs have set size $\Theta(K^2)$ or $\Theta(K^3)$ and flock size $K$. To the best of our knowledge, only a small amount of known families of periodic QCSSs with set size $\Theta(K^2)$ have been constructed and most of other known periodic QCSSs have set sizes much smaller than $\Theta(K^2)$. Our new constructed periodic QCSSs with set size $\Theta(K^2)$ and flock size $K$ have the best parameters among all known ones. They have larger set sizes or lower periodic tolerances. The periodic QCSSs with set size $\Theta(K^3)$ and flock size $K$ 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，准互补序列集被提出以支持多载波码分多址通信中更多的用户。在实际应用中，需要构造具有尽可能大的$M$和尽可能小的$\vartheta_{\max}$的$(M,K,N,\vartheta_{\max})$-准互补序列集，其中$M$表示集合中具有$K$行$N$列矩阵的数量，$\vartheta_{\max}$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准互补序列集一直是一个有趣的研究课题。迄今为止，文献中仅报道了少数渐近最优或接近最优的周期准互补序列集构造方法。本文基于有限域多项式和高斯和，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接近最优周期准互补序列集新族。这些准互补序列集族的集合规模为$\Theta(K^2)$或$\Theta(K^3)$，群集尺寸为$K$。据我们所知，目前仅构造了少量已知的集合规模为$\Theta(K^2)$的周期准互补序列集族，而大多数其他已知周期准互补序列集的集合规模远小于$\Theta(K^2)$。本文新构造的具有集合规模$\Theta(K^2)$和群集尺寸$K$的周期准互补序列集在所有已知序列集中具有最佳参数，它们具有更大的集合规模或更低的周期互相关容限。本文构造的具有集合规模$\Theta(K^3)$和群集尺寸$K$的周期准互补序列集在所有已知渐近最优或接近最优周期准互补序列集族中具有最大的集合规模。