  In this paper, a recent method to construct complementary sequence sets and
complete complementary codes by Hadamard matrices is deeply studied. By taking
the algebraic structure of Hadamard matrices into consideration, our main
result determine the so-called $\delta$-linear terms and $\delta$-quadratic
terms. As a first consequence, a powerful theory linking Golay complementary
sets of $p$-ary ($p$ prime) sequences and the generalized Reed-Muller codes by
Kasami et al. is developed. These codes enjoy good error-correcting capability,
tightly controlled PMEPR, and significantly extend the range of coding options
for applications of OFDM using $p^n$ subcarriers. As another consequence, we
make a previously unrecognized connection between the sequences in CSSs and
CCCs and the sequence with 2-level autocorrelation, trace function and
permutation polynomial (PP) over the finite fields.
