  A pure quantum state is called $k$-uniform if all its reductions to $k$-qudit
are maximally mixed. We investigate the general constructions of $k$-uniform
pure quantum states of $n$ subsystems with $d$ levels. We provide one
construction via symmetric matrices and the second one through classical
error-correcting codes. There are three main results arising from our
constructions. Firstly, we show that for any given even $n\ge 2$, there always
exists an $n/2$-uniform $n$-qudit quantum state of level $p$ for sufficiently
large prime $p$. Secondly, both constructions show that their exist $k$-uniform
$n$-qudit pure quantum states such that $k$ is proportional to $n$, i.e.,
$k=\Omega(n)$ although the construction from symmetric matrices outperforms the
one by error-correcting codes. Thirdly, our symmetric matrix construction
provides a positive answer to the open question in \cite{DA} on whether there
exists $3$-uniform $n$-qudit pure quantum state for all $n\ge 8$. In fact, we
can further prove that, for every $k$, there exists a constant $M_k$ such that
there exists a $k$-uniform $n$-qudit quantum state for all $n\ge M_k$. In
addition, by using concatenation of algebraic geometry codes, we give an
explicit construction of $k$-uniform quantum state when $k$ tends to infinity.
