Large Sets of Asymptotically Optimal and Near-Optimal Quasi-Complementary Sequences

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.