Hypercube Packings and Coverings with Higher Dimensional Rooks
Mehtaab Sawhney, David Stoner

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 -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.
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Taxonomy
TopicsCellular Automata and Applications · graph theory and CDMA systems · Coding theory and cryptography
