Classification of tight 2 s 2s 2 s-designs with s ≥ 2 s ≥ 2 s greater than or equals 2

Abstract

Tight 2 s $2 s$ 2 s -designs are the 2 s $2 s$ 2 s - ( v , k , λ ) (v, k, λ ) left parenthesis v comma k comma lamda right parenthesis designs whose sizes achieve the Fisher type lower bound ( v s ) v choose s StartBinomialOrMatrix v Choose s EndBinomialOrMatrix . Symmetric 2 $2$ 2 -designs, the Witt 4 $4$ 4 - ( 23 , 7 , 1 ) $(23, 7, 1)$ left parenthesis 23 comma 7 comma 1 right parenthesis design and the Witt 4 $4$ 4 - ( 23 , 16 , 52 ) $(23, 16, 52)$ left parenthesis 23 comma 16 comma 52 right parenthesis design are tight designs. It has been widely conjectured since the 1970s that there are no other nontrivial tight designs. In this paper, we give a proof of this conjecture. In the proof, an upper bound v ≪ s v ≪ s v much less than s is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound v ≫ s ( ln ⁡ s ) 2 v ≫ s (ln s)² v much greater than s left parenthesis ln s right parenthesis squared is shown by using estimates on prime gaps.

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BibTeX RIS

BibTeX

@article{xiang2026classification,
  title = {{Classification of tight 2 s $2s$ 2 s-designs with s ≥ 2 $s \geq 2$ s greater than or equals 2}},
  author = {Ziqing Xiang},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10287},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102874/type/journal_article},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Classification of tight 2 s 2s 2 s-designs with s ≥ 2 s ≥ 2 s greater than or equals 2
AU  - Ziqing Xiang
PY  - 2026
DA  - 2026/10/14
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10287
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102874/type/journal_article
AB  - Tight 2 s $2 s$ 2 s -designs are the 2 s $2 s$ 2 s - ( v , k , λ ) (v, k, λ ) left parenthesis v comma k comma lamda right parenthesis designs whose sizes achieve the Fisher type lower bound ( v s ) v choose s StartBinomialOrMatrix v Choose s EndBinomialOrMatrix . Symmetric 2 $2$ 2 -designs, the Witt 4 $4$ 4 - ( 23 , 7 , 1 ) $(23, 7, 1)$ left parenthesis 23 comma 7 comma 1 right parenthesis design and the Witt 4 $4$ 4 - ( 23 , 16 , 52 ) $(23, 16, 52)$ left parenthesis 23 comma 16 comma 52 right parenthesis design are tight designs. It has been widely conjectured since the 1970s that there are no other nontrivial tight designs. In this paper, we give a proof of this conjecture. In the proof, an upper bound v ≪ s v ≪ s v much less than s is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound v ≫ s ( ln ⁡ s ) 2 v ≫ s (ln s)^2 v much greater than s left parenthesis ln s right parenthesis squared is shown by using estimates on prime gaps.
SN  - 2050-5094
ER  -

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