Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

Bamberg, John; Lansdown, Jesse

doi:10.5802/alco.246

Bamberg, John ¹; Lansdown, Jesse ¹

¹ Centre for the Mathematics of Symmetry and Computation Department of Mathematics and Statistics The University of Western Australia, W.A., Australia

Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 197-212.

Abstract

In this paper we show that if $θ$ is a $T$ -design of an association scheme $(Ω, ℛ)$ , and the Krein parameters $q_{i, j}^{h}$ vanish for some $h \notin T$ and all $i, j \notin T$ ( $i, j, h \neq 0$ ), then $θ$ consists of precisely half of the vertices of $(Ω, ℛ)$ or it is a $T^{'}$ -design, where $| T^{'} | > | T |$ . We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$ -ovoids of generalised octagons of order $(s, s^{2})$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s, s^{2})$ ; (iii) the dual polar spaces $DQ (2 d, q)$ , $DW (2 d - 1, q)$ and $DH (2 d - 1, q^{2})$ , for $d \geq 3$ ; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in $Q^{-} (2 n - 1, q)$ , $n ⩾ 3$ .

Received: 2021-07-16
Revised: 2022-04-26
Accepted: 2022-05-05
Published online: 2023-02-24

DOI: 10.5802/alco.246

Classification: 05E30, 05B30, 05B25, 51E12, 51E05
Keywords: association schemes, Delsarte designs, Krein parameters, hemisystems, $m$-ovoids, generalised octagons, finite geometry

Author's affiliations:

Bamberg, John ¹; Lansdown, Jesse ¹

¹ Centre for the Mathematics of Symmetry and Computation Department of Mathematics and Statistics The University of Western Australia, W.A., Australia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2023__6_1_197_0,
     author = {Bamberg, John and Lansdown, Jesse},
     title = {Implications of vanishing {Krein} parameters on {Delsarte} designs, with applications in finite geometry},
     journal = {Algebraic Combinatorics},
     pages = {197--212},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {1},
     year = {2023},
     doi = {10.5802/alco.246},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.246/}
}

TY  - JOUR
AU  - Bamberg, John
AU  - Lansdown, Jesse
TI  - Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 197
EP  - 212
VL  - 6
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.246/
DO  - 10.5802/alco.246
LA  - en
ID  - ALCO_2023__6_1_197_0
ER  -

%0 Journal Article
%A Bamberg, John
%A Lansdown, Jesse
%T Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry
%J Algebraic Combinatorics
%D 2023
%P 197-212
%V 6
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.246/
%R 10.5802/alco.246
%G en
%F ALCO_2023__6_1_197_0

Bamberg, John; Lansdown, Jesse. Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 197-212. doi : 10.5802/alco.246. https://alco.centre-mersenne.org/articles/10.5802/alco.246/

References
Cited by

[1] Bamberg, John; Lansdown, Jesse; Lee, Melissa On $m$ -ovoids of regular near polygons, Des. Codes Cryptogr., Volume 86 (2018) no. 5, pp. 997-1006 | DOI | MR | Zbl

[2] Bamberg, John; Lee, Melissa A relative $m$ -cover of a Hermitian surface is a relative hemisystem, J. Algebraic Combin., Volume 45 (2017) no. 4, pp. 1217-1228 | DOI | MR | Zbl

[3] Bamberg, John; Metsch, Klaus On intriguing sets of the Penttila-Williford association scheme, Linear Algebra Appl., Volume 582 (2019), pp. 327-345 | DOI | MR | Zbl

[4] Bannai, Eiichi; Ito, Tatsuro Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984, xxiv+425 pages | MR

[5] Brouwer, A. E.; Cohen, A. M.; Neumaier, A. Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 18, Springer-Verlag, Berlin, 1989, xviii+495 pages | DOI | MR

[6] Cameron, P. J.; Goethals, J.-M.; Seidel, J. J. The Krein condition, spherical designs, Norton algebras and permutation groups, Nederl. Akad. Wetensch. Indag. Math., Volume 40 (1978) no. 2, pp. 196-206 | DOI | MR | Zbl

[7] Cameron, P. J.; Goethals, J.-M.; Seidel, J. J. Strongly regular graphs having strongly regular subconstituents, J. Algebra, Volume 55 (1978) no. 2, pp. 257-280 | DOI | MR | Zbl

[8] Cameron, P. J.; Liebler, R. A. Tactical decompositions and orbits of projective groups, Linear Algebra Appl., Volume 46 (1982), pp. 91-102 | DOI | MR | Zbl

[9] Cerzo, Diana R.; Suzuki, Hiroshi Non-existence of imprimitive $Q$ -polynomial schemes of exceptional type with $d = 4$ , European J. Combin., Volume 30 (2009) no. 3, pp. 674-681 | DOI | MR | Zbl

[10] Coolsaet, Kris; Jurišić, Aleksandar Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs, J. Combin. Theory Ser. A, Volume 115 (2008) no. 6, pp. 1086-1095 | DOI | MR | Zbl

[11] Cossidente, Antonio; Pavese, Francesco Relative $m$ -ovoids of elliptic quadrics, Discrete Math., Volume 342 (2019) no. 5, pp. 1481-1488 | DOI | MR | Zbl

[12] De Boeck, Maarten; Rodgers, Morgan; Storme, Leo; Švob, Andrea Cameron-Liebler sets of generators in finite classical polar spaces, J. Combin. Theory Ser. A, Volume 167 (2019), pp. 340-388 | DOI | MR | Zbl

[13] De Boeck, Maarten; Storme, Leo; Švob, Andrea The Cameron-Liebler problem for sets, Discrete Math., Volume 339 (2016) no. 2, pp. 470-474 | DOI | MR | Zbl

[14] Delsarte, P. An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. (1973) no. 10, p. vi+97 | MR | Zbl

[15] Feit, Walter; Higman, Graham The nonexistence of certain generalized polygons, J. Algebra, Volume 1 (1964), pp. 114-131 | DOI | MR | Zbl

[16] Filmus, Yuval; Ihringer, Ferdinand Boolean degree 1 functions on some classical association schemes, J. Combin. Theory Ser. A, Volume 162 (2019), pp. 241-270 | DOI | MR | Zbl

[17] Godsil, C. D. Krein covers of complete graphs, Australas. J. Combin., Volume 6 (1992), pp. 245-255 | MR | Zbl

[18] Jurišić, Aleksandar; Koolen, Jack Krein parameters and antipodal tight graphs with diameter 3 and 4, Discrete Math., Volume 244 (2002) no. 1-3, pp. 181-202 Algebraic and topological methods in graph theory (Lake Bled, 1999) | DOI | MR | Zbl

[19] Jurišić, Aleksandar; Vidali, Janoš Extremal 1-codes in distance-regular graphs of diameter 3, Des. Codes Cryptogr., Volume 65 (2012) no. 1-2, pp. 29-47 | DOI | MR | Zbl

[20] Makhnev, A. A. On the nonexistence of strongly regular graphs with the parameters $(486, 165, 36, 66)$ , Ukraïn. Mat. Zh., Volume 54 (2002) no. 7, pp. 941-949 | DOI | MR

[21] Martin, William J. Symmetric designs, sets with two intersection numbers and Krein parameters of incidence graphs, J. Combin. Math. Combin. Comput., Volume 38 (2001), pp. 185-196 | MR | Zbl

[22] Metsch, Klaus A gap result for Cameron-Liebler $k$ -classes, Discrete Math., Volume 340 (2017) no. 6, pp. 1311-1318 | DOI | MR | Zbl

[23] Östergård, Patric R. J.; Soicher, Leonard H. There is no McLaughlin geometry, J. Combin. Theory Ser. A, Volume 155 (2018), pp. 27-41 | DOI | MR | Zbl

[24] Penttila, Tim; Williford, Jason New families of $Q$ -polynomial association schemes, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 502-509 | DOI | MR | Zbl

[25] Rodgers, Morgan; Storme, Leo; Vansweevelt, Andries Cameron-Liebler $k$ -classes in $PG (2 k + 1, q)$ , Combinatorica, Volume 38 (2018) no. 3, pp. 739-757 | DOI | MR | Zbl

[26] Roos, Cornelis On antidesigns and designs in an association scheme, Delft Progr. Rep., Volume 2 (1982) no. 2, pp. 98-109 | MR | Zbl

[27] Smith, M. S. On rank $3$ permutation groups, J. Algebra, Volume 33 (1975), pp. 22-42 | DOI | MR | Zbl

[28] Suzuki, Hiroshi Imprimitive $Q$ -polynomial association schemes, J. Algebraic Combin., Volume 7 (1998) no. 2, pp. 165-180 | DOI | MR | Zbl

[29] Tanaka, Hajime; Tanaka, Rie Nonexistence of exceptional imprimitive $Q$ -polynomial association schemes with six classes, European J. Combin., Volume 32 (2011) no. 2, pp. 155-161 | DOI | MR | Zbl

[30] Terwilliger, Paul The subconstituent algebra of an association scheme. I, J. Algebraic Combin., Volume 1 (1992) no. 4, pp. 363-388 | DOI | MR | Zbl

[31] Thas, J. A. Interesting pointsets in generalized quadrangles and partial geometries, Linear Algebra Appl., Volume 114/115 (1989), pp. 103-131 | DOI | MR | Zbl

[32] Urlep, Matjaž Triple intersection numbers of $Q$ -polynomial distance-regular graphs, European J. Combin., Volume 33 (2012) no. 6, pp. 1246-1252 | DOI | MR | Zbl

[33] van Dam, Edwin R.; Koolen, Jack H.; Tanaka, Hajime Distance-regular graphs, Electron. J. Combin., Volume #DS22 (2016), p. 156 pages | MR | Zbl

[34] van Lint, J. H.; Wilson, R. M. A course in combinatorics, Cambridge University Press, Cambridge, 2001, xiv+602 pages | DOI | MR

[35] Vanhove, Frédéric Incidence geometry from an algebraic graph theory point of view, Ph. D. Thesis, Ghent University (2011)

[36] Vidali, Janoš Using symbolic computation to prove nonexistence of distance-regular graphs, Electron. J. Combin., Volume 25 (2018) no. 4, p. Paper 4.21, 10 pages | MR | Zbl

[37] Wolfram Research Inc. Mathematica, Version 12.0 (Champaign, IL, 2021)

Cited by Sources: