  Pure multipartite quantum states of n parties and local dimension q are
called k-uniform if all reductions to k parties are maximally mixed. These
states are relevant for our understanding of multipartite entanglement, quantum
information protocols, and the construction of quantum error correction codes.
To our knowledge, the only known systematic construction of these quantum
states is based on classical error correction codes. We present a systematic
method to construct other examples of k-uniform states and show that the states
derived through our construction are not equivalent to any k-uniform state
constructed from the so-called maximum distance separable error correction
codes. Furthermore, we use our method to construct several examples of
absolutely maximally entangled states whose existence was open so far.
