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Hypercube Packings and Coverings with Higher Dimensional Rooks

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Mehtaab Sawhney, David Stoner

Combinatorics · Submitted 31 Jan 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given these analogs, in the covering case we establish that the sphere-packing bound is asymptotically never tight except in trivial cases.

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arXiv ID1801.10607 (version 1)
Pith IDFWPFQGPY, Pith's permanent record of this paper and its review
CitationDavid Stoner, Mehtaab Sawhney. "Hypercube Packings and Coverings with Higher Dimensional Rooks." arXiv:1801.10607 (2018). https://pith.science/paper/FWPFQGPY
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,021 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.64 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.73 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 1.00 0.92 0.08 asserted 0.98
s1 More specifically, we generalize the notion of $1$-covering, $1$-packing, and $2$-packing in the case of $q$-ary codes. contribution 0.96 0.87 0.05 asserted 0.94
s2 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. result 0.79 0.95 0.18 asserted 1.00
s3 Given these analogs, in the covering case we establish that the sphere-packing bound is asymptotically never tight except in trivial cases. result 1.00 0.94 0.26 asserted 1.00
s4 This is in essence an analog of a seminal result of Rodemich regarding $q$-ary codes. interpretation 0.65 0.27 0.09 asserted 0.89
s5 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. result 0.97 0.94 0.28 speculative 0.59

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

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