Slepian-Wolf type problems on the erasure channel

Suppose that we have two users each of which has a k-dimensional vector over a field F_Q. Their goal is to communicate their vectors to a common receiver. At the time of reception, the receiver is given side information consisting of some of the entries of the first vector, some entries of the second vector, and the knowledge that some other entries of the two vectors are equal. The task is to design encoders for the two users in such a way that the receiver is able to recover the two vectors when given the side information and some of the entries of the encoded vectors. We call this problem the simultaneous erasure Slepian-Wolf problem, and we provide an optimal solution to this problem using rank-metric codes, provided that the field size.