Hypercube Packings and Coverings with Higher Dimensional Rooks

TL;DR

This paper generalizes classical q-ary codes by allowing points to cover or pack in higher dimensions with respect to Hamming distances 1 and 2, establishing bounds and asymptotic behaviors for these generalized codes.

Contribution

It introduces a new framework for q-ary codes with higher-dimensional rook moves, establishing sphere-packing and singleton bounds, and analyzing their tightness and asymptotic properties.

Findings

Sphere-packing bound is asymptotically not tight except in trivial cases.

Singleton bound analogs are tight in several cases for 1- and 2-packings.

Conjecture that these bounds are optimal in general.

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.