Nonexistence Results for Tight Block Designs

Jesse Short-Gershman; Peter Dukes

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Nonexistence Results for Tight Block Designs

Peter Dukes, Jesse Short-Gershman

Combinatorics · Submitted 16 Oct 2011

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

The authors' main claim Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,374 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.92 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.78 · compares with prior results 0.87 · mentions code or data availability 0.03.

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

Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, although no concrete understanding of `finitely many' is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$. Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight.	background 1.00	0.04	0.02	asserted 1.00
s1	A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, although no concrete understanding of `finitely many' is given.	background 0.69	0.03	0.06	asserted 1.00
s2	Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence.	method 0.92	0.89	0.06	asserted 1.00
s3	Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$.	result 0.79	0.97	0.61	asserted 0.98
s4	Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$.	result 0.89	0.95	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.59.

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Original abstract

Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, although no concrete understanding of `finitely many' is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$. Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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[ref] , [ref] , [ref] , [ref] , [ref] , etc. for tight spherical $t$-designs, and Ray-Chaudhuri and Wilson [ref] , Enomoto-Ito-Noda [ref] , Bannai [ref] , Dukes-ShortGershman [ref] , etc. for tight combinatorial designs.) For tight Euclidean $t$-designs, the theory was developed in certain cases ( [ref] ).

On the existence of tight relative 2-designs on binary Hamming association schemes · Introduction · 0.99

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