Two Constructions of Asymptotically Optimal Quasi-Complementary Sequence Sets

Abstract

An <inline-formula> <tex-math notation= LaTeX >$(M,K,N,\\delta _{\\mathrm {max}})$ </tex-math></inline-formula>-quasi-complementary sequence set (QCSS) is referred to as a set of <inline-formula> <tex-math notation= LaTeX >$M$ </tex-math></inline-formula> two-dimensional matrices of size <inline-formula> <tex-math notation= LaTeX >$K\\times N$ </tex-math></inline-formula> with maximum periodic correlation magnitude <inline-formula> <tex-math notation= LaTeX >$\\delta _{\\mathrm {max}}$ </tex-math></inline-formula>. It can be applied to a multi-carrier code-division multiple-access communication system to achieve low-interference performance. Compared with perfect complementary sequence sets, QCSSs with maximum periodic correlation magnitudes achieving or asymptotically achieving the correlation lower bounds have the advantage of supporting more users. In this paper, two constructions of periodic QCSSs from additive characters and multiplicative characters of finite fields are developed. In the first construction, new QCSSs with constituent sequence length <inline-formula> <tex-math notation= LaTeX >$N=p$ </tex-math></inline-formula> are proposed by using cyclic classes, where <inline-formula> <tex-math notation= LaTeX >$p$ </tex-math></inline-formula> is a prime. In the second construction, QCSSs with constituent sequence length <inline-formula> <tex-math notation= LaTeX >$N=q-1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$N=r^{2}-1$ </tex-math></inline-formula> are presented by employing almost difference sets and special sets proposed by Katz, respectively, where <inline-formula> <tex-math notation= LaTeX >$q>5$ </tex-math></inline-formula> is an odd prime power and <inline-formula> <tex-math notation= LaTeX >$r$ </tex-math></inline-formula> is a prime power. Notably, the parameters of QCSSs derived from the second construction are flexible and have not been covered in the literature. In addition, all the proposed periodic QCSSs are asymptotically optimal with respect to the correlation lower bounds.