Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and
minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at
the heart of coding theory. This work considers a generalization $n^\bx_q(M,d)$
of $n_q(M,d)$ corresponding to codes in which codewords have \emph{protected}
and \emph{unprotected} entries; where (analogs of) distance and of length are
measured with respect to protected entries only. Such codes, here referred to
as \emph{box codes}, have seen prior studies in the context of bipartite graph
covering. Upper and lower bounds on $n^\bx_q(M,d)$ are presented.