  A new method to construct $q$-ary complementary sequence sets (CSSs) and
complete complementary codes (CCCs) of size $N$ is proposed by using desired
para-unitary (PU) matrices. The concept of seed PU matrices is introduced and a
systematic approach on how to compute the explicit forms of the functions in
constructed CSSs and CCCs from the seed PU matrices is given. A general form of
these functions only depends on a basis of the functions from $\Z_N$ to $\Z_q$
and representatives in the equivalent class of Butson-type Hadamard (BH)
matrices. Especially, the realization of Golay pairs from the our general form
exactly coincides with the standard Golay pairs. The realization of ternary
complementary sequences of size $3$ is first reported here. For the realization
of the quaternary complementary sequences of size 4, almost all the sequences
derived here are never reported before. Generalized seed PU matrices and the
recursive constructions of the desired PU matrices are also studied, and a
large number of new constructions of CSSs and CCCs are given accordingly. From
the perspective of this paper, all the known results of CSSs and CCCs with
explicit GBF form in the literature (except non-standard Golay pairs) are
constructed from the Walsh matrices of order 2. This suggests that the proposed
method with the BH matrices of higher orders will yield a large number of new
CSSs and CCCs with the exponentially increasing number of the sequences of low
peak-to-mean envelope power ratio.
