Bounds on Box Codes

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 $n_q^{\text{bx}}(M,d)$ 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 $n_q^{\text{bx}}(M,d)$ are presented.