Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On classifying Minkowskian sublattices


Authors: Wolfgang Keller, Jacques Martinet and Achill Schürmann; with an Appendix by Mathieu Dutour Sikirić
Journal: Math. Comp. 81 (2012), 1063-1092
MSC (2010): Primary 11H55, 11H71
DOI: https://doi.org/10.1090/S0025-5718-2011-02528-7
Published electronically: September 12, 2011
Supplement 1: Explanation of supplementary material
Supplement 2: Appendix
MathSciNet review: 2869050
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BM09] A.-M. Bergé and J. Martinet, On perfection relations in lattices, in Quadratic Forms -- Algebra, Arithmetic, Geometry, Contemp. Math. 493 (2009), 29-49. MR 2537092 (2010f:11116)
  • [BCS95] A. Bonnecaze, P. Solé and A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41 (1995), 366-377. MR 1326285 (96b:94027)
  • [CE03] H. Cohn and N. Elkies, New upper bounds on sphere packings. I, Ann. Math. 157 (2003), 689-714. MR 1973059 (2004b:11096)
  • [CK09] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. Math. 170 (2009), 1003-1050. MR 2600869
  • [CS99] J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Springer, New York, 1999, 3rd ed. MR 1662447 (2000b:11077)
  • [DSV07] M. Dutour Sikirić, A. Schürmann, and F. Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21-32. MR 2300003 (2007m:11089)
  • [Mar01] J. Martinet (with an appendix by Christian Batut), Sur l'indice d'un sous-réseau, Réseaux Euclidiens designs sphériques et forms modulaires, Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001, pp. 163-211. MR 1878750 (2002k:11109)
  • [Mar02] J. Martinet, Reduction modulo 2 and 3 of Euclidean lattices, J. Algebra 251 (2002), 864-887. MR 1919157 (2003e:11074)
  • [Mar03] -, Perfect lattices in Euclidean spaces, Springer, Berlin, 2003. MR 1957723 (2003m:11099)
  • [MS10] J. Martinet and A. Schürmann, Bases of minimal vectors in lattices III, in preparation.
  • [McK98] B. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324. MR 1606516 (98k:68132)
  • [Ryš76] S.S. Ryškov (=Ryshkov) , On the problem of determining perfect quadratic forms of several variables (in Russian), Trudy Mat. Inst. Steklov. 142 (1976), 215-239, 270-271, English translation by the AMS, 1979, pp. 233-259. MR 0563098 (58:27807)
  • [Sch09a] A. Schürmann, Computational geometry of positive definite quadratic forms, University Lecture Series 49, AMS, Providence, 2009. MR 2466406 (2010a:11130)
  • [Sch09b] -, Enumerating perfect forms, in Quadratic Forms -- Algebra, Arithmetic, Geometry, Contemp. Math. 493 (2009), 359-377. MR 2537111 (2010g:11110)
  • [Ser03] A. Seress, Permutation group algorithms, Cambridge University Press, 2003. MR 1970241 (2004c:20008)
  • [Vor07] 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.
  • [Wat71a] G.L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math. Rozprawy Mat. 84 (1971), 42 pp. MR 0318061 (47:6610)
  • [Wat71b] -, On the minimum points of a positive quadratic form, Mathematika 18 (1971), 60-70. MR 0289421 (44:6612)
  • [Zah80] N.V. Zaharova, Centerings of eight-dimensional lattices that preserve a frame of successive minima, Trudy Mat. Inst. Steklov. 152 (1980), 97-123, 237, English translation by the AMS, 1982, pp. 107-134. Maxim Anzin, in an e-mail dated March 23rd, 2004, pointed out to the second author that the three possible structures which were forgotten in [Zah80] (quoted in [Mar01]) were corrected in a preprint in Russian written under the name of N. V. Novikova, a preprint that we have never seen. MR 603817 (82k:10033)
  • [Zie97] G.M. Ziegler, Lectures on polytopes, Graduate Texts in Math., 152, Springer, New York, 1995. MR 1311028 (96a:52011)
    SOFTWARE
  • [Convex] Convex - a Maple package for convex geometry, ver. 1.1.3, by Matthias Franz, http://www.math.uwo.ca/~mfranz/convex/.
  • [lrs] 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] 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] Polyhedral - a GAP package, by M. Dutour Sikirić, http://www.liga.ens.fr/~dutour/Polyhedral/index.html.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11H55, 11H71

Retrieve articles in all journals with MSC (2010): 11H55, 11H71


Additional Information

Wolfgang Keller
Affiliation: Faculty of Mathematics, Otto-von-Guericke Universität, 39106 Magdeburg, Germany
Email: Wolfgang.Keller@student.uni-magdeburg.de

Jacques Martinet
Affiliation: Institut de Mathématiques, 351, cours de la Libération, 33405 Talence cedex, France
Email: Jacques.Martinet@math.u-bordeaux1.fr

Achill Schürmann
Affiliation: Institute of Mathematics, University of Rostock, 18051 Rostock, Germany
Email: achill.schuermann@uni-rostock.de

Mathieu Dutour Sikirić
Affiliation: Rudjer Bosković Institute, Bijenicka 54, 10000 Zagreb, Croatia
Email: mdsikir@irb.hr

DOI: https://doi.org/10.1090/S0025-5718-2011-02528-7
Keywords: Euclidean lattices, quadratic forms, linear codes
Received by editor(s): April 20, 2009
Received by editor(s) in revised form: January 29, 2011
Published electronically: September 12, 2011
Additional Notes: The first and the third authors were supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4-2. The third author was additionally supported by the Université Bordeaux 1
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society