  In this work, we construct $4$-phase Golay complementary sequence (GCS) set
of cardinality $2^{3+\lceil \log_2 r \rceil}$ with arbitrary sequence length
$n$, where the $10^{13}$-base expansion of $n$ has $r$ nonzero digits.
Specifically, the GCS octets (eight sequences) cover all the lengths no greater
than $10^{13}$. Besides, based on the representation theory of signed symmetric
group, we construct Hadamard matrices from some special GCS to improve their
asymptotic existence: there exist Hadamard matrices of order $2^t m$ for any
odd number $m$, where $t = 6\lfloor \frac{1}{40}\log_{2}m\rfloor + 10$.
