  We investigate the maximum purity that can be achieved by k-uniform mixed
states of N parties. Such N-party states have the property that all their
k-party reduced states are maximally mixed. A scheme to construct explicitly
k-uniform states using a set of specific N-qubit Pauli matrices is proposed. We
provide several different examples of such states and demonstrate that in some
cases the state corresponds to a particular orthogonal array. The obtained
states, despite being mixed, reveal strong non-classical properties such as
genuine multipartite entanglement or violation of Bell inequalities.
