2000 character limit reached
Classification of tight $2s$-designs with $s \geq 2$ (2312.14778v1)
Published 22 Dec 2023 in math.CO
Abstract: Tight $2 s$-designs are the $2 s$-$(v, k, \lambda)$ designs whose sizes achieve the Fisher type lower bound ${v \choose s}$. Symmetric $2$-designs, the Witt $4$-$(23, 7, 1)$ design and the Witt $4$-$(23, 16, 52)$ design are tight designs. It has been widely conjectured since 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 $v \ll s$ is shown by analyzing the parameters of the designs and the coefficients of the Wilson polynomials, and a lower bound $v \gg s (\ln s)2$ is shown by using estimates on prime gaps.
- R.C. Baker, G. Harman and J. Pintz “The difference between consecutive primes. II” In Proceedings of the London Mathematical Society. Third Series 83.3, 2001, pp. 532–562 DOI: 10.1112/plms/83.3.532
- Eiichi Bannai “On tight designs” In The Quarterly Journal of Mathematics. Oxford Second Series 28, 1977, pp. 433–448 DOI: 10.1093/qmath/28.4.433
- Andrew Bremner “A Diophantine equation arising from tight 4-designs” In Osaka Journal of Mathematics 16, 1979, pp. 353–356 DOI: 10.18910/8744
- H. Cramér “On the order of magnitude of the difference between consecutive prime numbers” In Acta Arithmetica 2, 1936, pp. 23–46 DOI: 10.4064/aa-2-1-23-46
- P. Delsarte “An algebraic approach to the association schemes of coding theory.” Ann Arbor, MI: Historical Jrl., 1973
- “Nonexistence results for tight block designs” In Journal of Algebraic Combinatorics 38.1, 2013, pp. 103–119 DOI: 10.1007/s10801-012-0395-8
- Pierre Dusart “Explicit estimates of some functions over primes” In The Ramanujan Journal 45.1, 2018, pp. 227–251 DOI: 10.1007/s11139-016-9839-4
- Hikoe Enomoto, Noboru Ito and Ryuzaburo Noda “Tight 4-designs” In Osaka Journal of Mathematics 16, 1979, pp. 39–43 DOI: 10.18910/7614
- R.A. Fisher “An examination of the different possible solutions of a problem in incomplete blocks” In Annals of Eugenics 10, 1940 DOI: 10.1111/j.1469-1809.1940.tb02237.x
- Peter Keevash “The existence of designs”, 2014 DOI: 10.48550/arxiv.1401.3665
- Cheryl Peterson “On tight 6-designs” In Osaka Journal of Mathematics 14, 1977, pp. 417–435 DOI: 10.18910/3990
- A.Ya. Petrenyuk “Fisher’s inequality for tactical configurations” In Matematicheskie Zametki 4, 1968, pp. 417–424 URL: https://www.mathnet.ru/eng/mzm9463
- Dijen K. Ray-Chaudhuri and Richard M. Wilson “On t-designs” In Osaka Journal of Mathematics 12, 1975, pp. 737–744 DOI: 10.18910/7296
- Roelof J. Stroeker “On the Diophantine equations (2y2−3)2=x2(3x2−2)superscript2superscript𝑦232superscript𝑥23superscript𝑥22(2y^{2}-3)^{2}=x^{2}(3x^{2}-2)( 2 italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 3 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 3 italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 ) in connection with the existence of non-trivial tight 4-designs” In Indagationes Mathematicae 43, 1981, pp. 353–358 DOI: 10.1016/1385-7258(81)90046-9
- Luc Teirlinck “Non-trivial t𝑡titalic_t-designs without repeated blocks exist for all t𝑡titalic_t” In Discrete Mathematics 65, 1987, pp. 301–311 DOI: 10.1016/0012-365X(87)90061-6
- Aimo Tietäväinen “On the nonexistence of perfect codes over finite fields” In SIAM J. Appl. Math. 24, 88-96 (1973)., 1972 DOI: 10.1137/0124010
- Richard M. Wilson “An existence theory for pairwise balanced designs. I: Composition theorems and morphisms” In Journal of Combinatorial Theory. Series A 13, 1972, pp. 220–245 DOI: 10.1016/0097-3165(72)90028-3
- Richard M. Wilson “An existence theory for pairwise balanced designs. II: Structure of PBD- closed sets and the existence conjectures” In Journal of Combinatorial Theory. Series A 13, 1972, pp. 246–273 DOI: 10.1016/0097-3165(72)90029-5
- Richard M. Wilson “An existence theory for pairwise balanced designs. III: Proof of the existence conjectures” In Journal of Combinatorial Theory. Series A 18, 1975, pp. 71–79 DOI: 10.1016/0097-3165(75)90067-9
- Ziqing Xiang “Nonexistence of nontrivial tight 8-designs” In Journal of Algebraic Combinatorics 47.2, 2018, pp. 301–318 DOI: 10.1007/s10801-017-0776-0
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.