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Group divisible designs with block size 4 and type g^u m^1 - II

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Anthony D. Forbes

Combinatorics · Submitted 19 Jun 2018

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

The authors' main claim We show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35.

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arXiv ID1806.07491 (version 1)
Pith IDHSGXZE2P, Pith's permanent record of this paper and its review
CitationAnthony D. Forbes. "Group divisible designs with block size 4 and type g^u m^1 - II." arXiv:1806.07491 (2018). https://pith.science/paper/HSGXZE2P
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,933 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.78 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.73 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for all g congruent to 3 (mod 6), the possible exceptions occur only when u = 8 and g is not divisible by any of 9, 15, 21, 33, 39, 51, 57, 69, 87 or 93. Consequently we are able to extend the known spectrum for g congruent to 1 and g congruent to 5 (mod 6). Also we complete the spectrum for 4-GDDs of type (3a)^4 (6a)^1 (3b)^1.

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 show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. result 0.91 0.95 0.68 asserted 1.00
s1 More generally, we show that for all g congruent to 3 (mod 6), the possible exceptions occur only when u = 8 and g is not divisible by any of 9, 15, 21, 33, 39, 51, 57, 69, 87 or 93. result 0.97 0.95 0.33 asserted 0.98
s2 Consequently we are able to extend the known spectrum for g congruent to 1 and g congruent to 5 (mod 6). interpretation 0.73 0.90 0.13 asserted 1.00
s3 Also we complete the spectrum for 4-GDDs of type (3a)^4 (6a)^1 (3b)^1. result 0.80 0.95 0.09 asserted 1.00

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

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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 show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for all g congruent to 3 (mod 6), the possible exceptions occur only when u = 8 and g is not divisible by any of 9, 15, 21, 33, 39, 51, 57, 69, 87 or 93. Consequently we are able to extend the known spectrum for g congruent to 1 and g congruent to 5 (mod 6). Also we complete the spectrum for 4-GDDs of type (3a)^4 (6a)^1 (3b)^1.

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