On classifying Minkowskian sublattices

Keller, Wolfgang; Martinet, Jacques; Schürmann, Achill

doi:10.1090/S0025-5718-2011-02528-7

On classifying Minkowskian sublattices
HTML articles powered by AMS MathViewer

by Wolfgang Keller, Jacques Martinet and Achill Schürmann; with an Appendix by Mathieu Dutour Sikirić PDF

Math. Comp. 81 (2012), 1063-1092 Request permission

Abstract:

Let $\Lambda$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lambda ’$ be a Minkowskian sublattice of $\Lambda$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lambda$. We extend the classification of possible $\mathbb {Z}/d\mathbb {Z}$-codes of the quotients $\Lambda /\Lambda ’$ to dimension $9$, where $d\mathbb {Z}$ is the annihilator of $\Lambda /\Lambda ’$.

References

Anne-Marie Bergé and Jacques Martinet, On perfection relations in lattices, Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 29–49. MR 2537092, DOI 10.1090/conm/493/09664
Alexis Bonnecaze, Patrick Solé, and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41 (1995), no. 2, 366–377. MR 1326285, DOI 10.1109/18.370138
Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, DOI 10.4007/annals.2003.157.689
Henry Cohn and Abhinav Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2) 170 (2009), no. 3, 1003–1050. MR 2600869, DOI 10.4007/annals.2009.170.1003
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
Mathieu Dutour Sikirić, Achill Schürmann, and Frank Vallentin, Classification of eight-dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21–32. MR 2300003, DOI 10.1090/S1079-6762-07-00171-0
Jacques Martinet, Sur l’indice d’un sous-réseau, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001, pp. 163–211 (French, with English and French summaries). With an appendix by Christian Batut. MR 1878750
Jacques Martinet, Reduction modulo 2 and 3 of Euclidean lattices, J. Algebra 251 (2002), no. 2, 864–887. MR 1919157, DOI 10.1006/jabr.2001.9018
Jacques Martinet, Perfect lattices in Euclidean spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Springer-Verlag, Berlin, 2003. MR 1957723, DOI 10.1007/978-3-662-05167-2
J. Martinet and A. Schürmann, Bases of minimal vectors in lattices III, in preparation.
Brendan D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (1998), no. 2, 306–324. MR 1606516, DOI 10.1006/jagm.1997.0898
S. S. Ryškov, On the problem of determining perfect quadratic forms of several variables, Trudy Mat. Inst. Steklov. 142 (1976), 215–239, 270–271 (Russian). Number theory, mathematical analysis and their applications. MR 0563098
Achill Schürmann, Computational geometry of positive definite quadratic forms, University Lecture Series, vol. 48, American Mathematical Society, Providence, RI, 2009. Polyhedral reduction theories, algorithms, and applications. MR 2466406, DOI 10.1090/ulect/048
Achill Schürmann, Enumerating perfect forms, Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 359–377. MR 2537111, DOI 10.1090/conm/493/09679
Ákos Seress, Permutation group algorithms, Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. MR 1970241, DOI 10.1017/CBO9780511546549
G.F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire. Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math. 133 (1907), 97–178.
G. L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math. (Rozprawy Mat.) 84 (1971), 42. MR 318061
G. L. Watson, On the minimum points of a postive quadratic form, Mathematika 18 (1971), 60–70. MR 289421, DOI 10.1112/S0025579300008378
N. V. Zaharova, Centerings of eight-dimensional lattices that preserve a frame of successive minima, Trudy Mat. Inst. Steklov. 152 (1980), 97–123, 237 (Russian). Geometry of positive quadratic forms. MR 603817
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1
Convex – a Maple package for convex geometry, ver. 1.1.3, by Matthias Franz, http://www.math.uwo.ca/~mfranz/convex/.
lrs – C implementation of the reverse search algorithm for vertex enumeration, ver. 4.2, by D. Avis, http://cgm.cs.mcgill.ca/~avis/C/lrs.html.
MAGMA – high performance software for Algebra, Number Theory, and Geometry, ver. 2.13, by the Computational Algebra Group at the University of Sydney, http://magma.maths.usyd.edu.au/.
Polyhedral – a GAP package, by M. Dutour Sikirić, http://www.liga.ens.fr/~dutour/Polyhedral/index.html.