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 $n_{q}^{\bullet}(M, d)$ of $n_{q}(M, d)$ corresponding to codes in which codewords have protected and unprotected entries; where (analogs of) distance and of length are measured with respect to protected entries only. Such …