  The correlation properties of sequences form a focal point in the design of
multiple access systems of communications. Such a system must be able to serve
a number of simultaneous users while keeping interference low. A popular choice
for the set of sequences to deploy is the quasi-complementary sequence set
(QCSS). Its large set size enables the system to accommodate a lot of users.
The set has low nontrivial correlation magnitudes within a zone around the
origin. This keeps undue interference among users under control. A QCSS
performs better than the perfect complementary sequence set (PCSS) does in
schemes with fractional delays.
  The optimality of a set of periodic sequences is measured by its maximum
periodic correlation magnitude, for which there is an established lower bound
to aim at. For a fixed period, optimal sets are known only for very restricted
parameters. Efforts have therefore been centered around the constructions of
asymptotically optimal sets. Their periods are allowed to be as large as
sufficient to establish optimality. In this paper we share an insight that a
sequence set that asymptotically attains the Welch bound generates an
asymptotically optimal periodic QCSS by interleaving. One can simply use known
families of such sequence sets to construct the desired QCSSs. Seven families
of QCSSs with specific parameters are shown as examples of this general
construction. We build upon the insight to propose two new direct constructions
of asymptotically optimal QCSSs with very flexible parameters without
interleaving. The flexibility enhances their appeal for practical
implementation. The mathematical tools come from the theory of groups in the
form of additive and multiplicative characters of finite fields.
