  Genuinely entangled subspaces (GESs) are the class of completely entangled
subspaces that contain only genuinely multiparty entangled states. They
constitute a particularly useful notion in the theory of entanglement but also
have found an application, for instance, in quantum error correction and
cryptography. In a recent study (Demianowicz and Augusiak in Phys Rev A
98:012313, 2018), we have shown how GESs can be efficiently constructed in any
multiparty scenario from the so-called unextendible product bases. The provided
subspaces, however, are not of maximal allowable dimensions, and our aim here
is to put forward an approach to building such. The method is illustrated with
few examples in small systems. Connections with other mathematical problems,
such as spaces of matrices of equal rank and the numerical range, are
discussed.
