Hypercube Packings and Coverings with Higher Dimensional Rooks
Abstract
We introduce a generalization of classical -ary codes by allowing points to cover other points that are Hamming distance or in a freely chosen subset of all directions. More specifically, we generalize the notion of -covering, -packing, and -packing in the case of -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 -ary codes. In the packing case we establish for the -packing and -packing cases that the analog of the singleton bound is tight in several possible cases and conjecture that these bounds are optimal in general.
Cite
@article{arxiv.1801.10607,
title = {Hypercube Packings and Coverings with Higher Dimensional Rooks},
author = {Mehtaab Sawhney and David Stoner},
journal= {arXiv preprint arXiv:1801.10607},
year = {2018}
}