  Golay complementary sequences have been put a high value on the applications
in orthogonal frequency-division multiplexing (OFDM) systems since its good
peak-to-mean envelope power ratio(PMEPR) properties. However, with the increase
of the code length, the code rate of the standard Golay sequences suffer a
dramatic decline. Even though a lot of efforts have been paid to solve the code
rate problem for OFDM application, how to construct large classes of sequences
with low PMEPR is still difficult and open now. In this paper, we propose a new
method to construct $q$-ary Golay complementary set of size $N$ and length
$N^n$ by $N\times N$ Hadamard Matrices where $n$ is arbitrary and $N$ is a
power of 2. Every item of the constructed sequences can be presented as the
product of the specific entries of the Hadamard Matrices. The previous works in
\cite{BudIT} can be regarded as a special case of the constructions in this
paper and we also obtained new quaternary Golay sets never reported in the
literature.
