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

Constructions of k-uniform and absolutely maximally entangled states beyond maximum distance codes