La notion de designs dans les espaces Grassmanniens a été introduite par l’auteur et R. Coulangeon, G. Nebe dans [3]. Après avoir rappelé les premières propriétés de ces objets et les relations avec la théorie des réseaux, nous montrons que la famille des réseaux de Barnes-Wall contient des -designs grassmanniens. Nous discutons également des relations entre cette notion de designs et les designs associés à l’espace symétrique formé des espaces totalement isotropes d’un espace quadratique binaire, qui sont mises en évidence par une certaine construction utilisant le groupe de Clifford.
The notion of designs in Grassmannian spaces was introduced by the author and R. Coulangeon, G. Nebe, in [3]. After having recalled some basic properties of these objects and the connections with the theory of lattices, we prove that the sequence of Barnes-Wall lattices hold -Grassmannian designs. We also discuss the connections between the notion of Grassmannian design and the notion of design associated with the symmetric space of the totally isotropic subspaces in a binary quadratic space, which is revealed in a certain construction involving the Clifford group.
@article{JTNB_2005__17_1_25_0,
author = {Christine Bachoc},
title = {Designs, groups and lattices},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {25--44},
publisher = {Universit\'e Bordeaux 1},
volume = {17},
number = {1},
year = {2005},
doi = {10.5802/jtnb.474},
zbl = {1074.05023},
mrnumber = {2152208},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.474/}
}
Christine Bachoc. Designs, groups and lattices. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 25-44. doi : 10.5802/jtnb.474. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.474/
[1] C. Bachoc, E. Bannai, R. Coulangeon, Codes and designs in Grassmannian spaces. Discrete Mathematics 277 (2004), 15–28. | MR 2033722 | Zbl 1040.05005
[2] C. Bachoc, Linear programming bounds for codes in Grassmannian spaces. In preparation.
[3] C. Bachoc, R. Coulangeon, G. Nebe, Designs in Grassmannian spaces and lattices. J. Algebraic Combinatorics 16 (2002), 5–19. | MR 1941981 | Zbl 1035.05027
[4] C. Bachoc, G. Nebe, Siegel modular forms, Grassmannian designs, and unimodular lattices. Proceedings of the 19th Algebraic Combinatorics Symposium, Kumamoto (2002).
[5] C. Bachoc, B. Venkov, Modular forms, lattices and spherical designs. In “Réseaux euclidiens, designs sphériques et formes modulaires”, J. Martinet, éd., L’Enseignement Mathématique, Monographie no 37, Genève (2001), 87–111. | MR 1878746 | Zbl 1061.11035
[6] E. Bannai, T. Ito, Agebraic Combinatorics I, Association Schemes (1984). | MR 882540 | Zbl 0555.05019
[7] M. Broué, M. Enguehard, Une famille infinie de formes quadratiques entières et leurs groupes d’automorphismes. Ann. Scient. E.N.S., série, 6 (1973), 17–52. | Numdam | MR 335654 | Zbl 0261.20022
[8] B. Bolt, The Clifford collineation, transform and similarity groups III: generators and involutions. J. Australian Math. Soc, 2 (1961), 334–344. | MR 142666 | Zbl 0107.02701
[9] B. Bolt, T.G. Room, G.E. Wall, On Clifford collineation, transform and similarity groups I. J. Australian Math. Soc, 2 (1961), 60–79 | MR 125874 | Zbl 0097.01702
[10] B. Bolt, T.G. Room, G.E. Wall, On Clifford collineation, transform and similarity groups II. J. Australian Math. Soc, 2 (1961), 80–96 | MR 125874 | Zbl 0097.01702
[11] J. H. Conway, R. H. Hardin, E. Rains, P.W. Shor, N. J. A. Sloane, A group-theoretical framework for the construction of packings in Grassmannian spaces. J. Algebraic Comb. 9 (1999), 129–140. | MR 1679247 | Zbl 0941.51033
[12] J. H. Conway, R. H. Hardin, N. J. A. Sloane, Packing Lines, Planes, etc., Packings in Grassmannian Spaces. Experimental Mathematics 5 (1996), 139–159. | MR 1418961 | Zbl 0864.51012
[13] R. Coulangeon, Réseaux -extrêmes. Proc. London Math. Soc. (3) 73 (1996), no. 3, 555–574. | MR 1407461 | Zbl 0861.11040
[14] P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs. Geom. Dedicata 6 (1977), 363–388. | MR 485471 | Zbl 0376.05015
[15] P. Delsarte, V.I. Levenshtein, Association schemes and coding theory. IEEE Trans. Inf. Th. 44 (6) (1998), 2477–2504. | MR 1658771 | Zbl 0946.05086
[16] R. Goodman, N. R. Wallach, Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, 1998. | MR 1606831 | Zbl 0901.22001
[17] A.T. James, A.G. Constantine, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold. Proc. London Math. Soc. (3) 29 (1974), 174–192. | MR 374523 | Zbl 0289.33031
[18] J. Martinet, Sur certains designs sphériques liés à des réseaux entiers. In “Réseaux euclidiens, designs sphériques et formes modulaires, J. Martinet, éd., L’Enseignement Mathématique, Monographie no 37, Genève (2001). | MR 1878748 | Zbl 1065.11050
[19] G. Nebe, W. Plesken, Finite rational matrix groups. Memoirs of the AMS, vol. 116, nb. 556 (1995). | MR 1265024 | Zbl 0837.20056
[20] G. Nebe, E. Rains, N.J.A Sloane, The invariants of the Clifford groups. Designs, Codes, and Cryptography 24 (1) (2001), 99–122. | MR 1845897 | Zbl 1002.11057
[21] G. Nebe, N.J.A Sloane, A catalogue of lattices. http://www.research.att.com/njas/lattices/index.html
[22] G. Nebe, B. Venkov, The strongly perfect lattices of dimension . J. Théorie de Nombres de Bordeaux 12 (2000), 503–518. | Numdam | MR 1823200 | Zbl 0997.11049
[23] G. Nebe, B. Venkov, The strongly perfect lattices of dimension . In preparation. | Numdam
[24] B. Runge, On Siegel modular forms I. J. Reine Angew. Math. 436 (1993), 57–85. | MR 1207281 | Zbl 0772.11015
[25] B. Runge, On Siegel modular forms II. Nagoya Math. J. 138 (1995), 179–197. | MR 1339948 | Zbl 0824.11031
[26] B. Runge, Codes and Siegel modular forms. Discrete Math. 148 (1995), 175–205. | MR 1368288 | Zbl 0854.11071
[27] D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups. Amer. J. Math. 102 (4) (1980), 625–662. | MR 584464 | Zbl 0448.33019
[28] D. Stanton, Orthogonal polynomials and Chevalley groups. In Special functions: Group theoretical aspects and applications R.A. Askey, T.H. Koornwinder, W. Schempp editors, Mathematics and its applications, D. Reidel Publishing Company, 1984. | MR 774056 | Zbl 0578.20041
[29] B. Venkov, Réseaux et designs sphériques. In “Réseaux euclidiens, designs sphériques et formes modulaires, J. Martinet, éd., L’Enseignement Mathématique, Monographie no 37, Genève (2001). | MR 1878745 | Zbl 1054.11034
[30] H. Wei, Y. Wang, Suborbits of the transitive set of subspaces of type (m,0) under finite classical groups. Algebra Colloq. 3:1 (1996), 73–84. | MR 1374163 | Zbl 0843.05009