  Inspired by problems in Private Information Retrieval, we consider the
setting where two users need to establish a communication protocol to transmit
a secret without revealing it to external observers. This is a question of how
large a linear code can be, when it is required to agree with a prescribed code
on a collection of coordinate sets. We show how the efficiency of such a
protocol is determined by the derived matroid of the underlying linear
communication code. Furthermore, we provide several sufficient combinatorial
conditions for when no secret transmission is possible.
