English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce a generalization of classical $q$-ary codes by allowing pointsto cover other points that are Hamming distance $1$ or $2$ in a freely chosensubset 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 thecovering case, we establish the analog of the sphere-packing bound and in thepacking case, we establish an analog of the singleton bound. Given theseanalogs, in the covering case we establish that the sphere-packing bound isasymptotically never tight except in trivial cases. This is in essence ananalog of a seminal result of Rodemich regarding $q$-ary codes. In the packingcase we establish for the $1$-packing and $2$-packing cases that the analog ofthe singleton bound is tight in several possible cases and conjecture thatthese bounds are optimal in general.

Hypercube Packings and Coverings with Higher Dimensional Rooks