Hypercube Packings and Coverings with Higher Dimensional Rooks

Abstract: We introduce a generalization of classical $q$-ary codes by allowing points to cover other points that are Hamming distance $1$ or $2$ in a freely chosen subset of all directions. More specifically, we generalize the notion of $1$-covering, $1$-packing, and $2$-packing in the case of $q$-ary codes. In the covering case, we establish the analog of the sphere-packing bound and in the packing case, we establish an analog of the singleton bound. Given these analogs, in the covering case we establish that the sphere-packing bound is asymptotically never tight except in trivial cases. This is in essence an analog of a seminal result of Rodemich regarding $q$-ary codes. In the packing case we establish for the $1$-packing and $2$-packing cases that the analog of the singleton bound is tight in several possible cases and conjecture that these bounds are optimal in general.