The problem of constructing a minimal rank matrix over $GF(2)$ whose kernel does not intersect a given set $S$ is considered.  In the case where $S$ is a Hamming ball centered at $0$, this is equivalent to finding linear codes of largest dimension, with the Gilbert-Varshamov entropy bound conjectured to be tight. Motivated by the problem of linear Boolean classification, this work focuses on the case of annulus. Characterizations of the minimal rank are given in various regimes and a general conjecture is proposed.