Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let Fpmbe the finite field withpmelements for a primepand a positive integerm. LetV(p)nbe ann-dimensional vector space over Fpfor a primep. In this paper, we study the hybrid character sums of the form Σx∈V(p)nψ (F(x)) χa(x), whereFis a function fromV(p)nto Fpm,a∈V(p)n, ψ is a nontrivial multiplicative character of Fpm, χa(x) = ζ⟨a,x⟩npis the character ofV(p)nand ⟨, ⟩ndenotes a (non-degenerate) inner product ofV(p)n. IfF(x) is a vectorial dual-bent function anda∈V(p)n\ {0}, we determine their complex modulus or explicit values under certain conditions. This generalizes some known results as special cases. We show that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The codebooks we construct have very small alphabet sizes. This enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.

In recent years, quasi-complementary sequence sets (QCSSs) have attracted widespread attention as they can support more users in MC-CDMA communications than perfect complementary sequence sets (PCSSs). The objective of this paper is to present three novel constructions of asymptotically optimal or near-optimal periodic QCSSs based on algebraic methods. Firstly, we propose a generic constriction of QCSSs with small alphabet sizepfrom polynomials over finite fields. Using the quadratic and cubic polynomials, we then respectively derive an infinite family of asymptotically optimal QCSSs and an infinite family of asymptotically near-optimal periodic QCSSs with large set sizes. Secondly, we give a construction of periodic QCSSs based on Gaussian sums which have smaller periodic tolerance than that of a known family of QCSSs. Thirdly, we present a construction of periodic QCSSs from permutation polynomials and complementary sets, yielding an infinite family of QCSSs with large set size, small periodic tolerance and low column sequence peak-to-average power ratio (PAPR).

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 (p2n,pn− 1,pn− 1,pn+ 1) QCSSs with small alphabet sizep, which has larger set size than the known family of periodic (pn(pn−1),pn−1,pn−1,pn+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 Θ(K2) for some flock sizeK. Compared with known asymptotically optimal aperiodic QCSSs in the literature, our proposed aperiodic QCSSs have better or new parameters. Particularly, for three families of the costructed aperiodic QCSSs, the column sequence peak-to-average power ratio (PAPR) is upper bounded by p if we select suitable column orthogonal complex matrices.