Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

Author(s): Kanat Abdukhalikov; Rudolf Scharlau.
Journal: Math. Comp.
MSC (2000): Primary 11H06, 11H56; Secondary 11E39, 11H71, 11F11
Posted: May 16, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $ \mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.

References:

1.
K. Abdukhalikov, Invariant Hermitian lattices in the Steinberg module and their isometry groups, Comm. Algebra 25 (8) (1997) 2607-2626. MR 1459581 (98e:20005)

2.
K. Abdukhalikov, Unimodular Hermitian lattices in dimension 13, J. Algebra 272 (1) (2004) 186-190. MR 2029031 (2005b:11098)

3.
C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A 78 (1997) 92-119. MR 1439633 (98a:11084)

4.
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (3/4) (1997) 235-265. MR 1484478

5.
A. M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379-436. MR 0422448 (54:10437)

6.
J. H. Conway, N. J. A. Sloane, Self-dual codes over $ GF(3)$ and $ GF(4)$ of length not exceeding $ 16$, IEEE Trans. Inform. Theory 25 (1979) 312-322. MR 528009 (80h:94026)

7.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, ATLAS of finite groups, Clarendon Press, Oxford, 1985. MR 827219 (88g:20025)

8.
J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Springer-Verlag New York, Inc., 1999. MR 1662447 (2000b:11077)

9.
R. Coulangeon, Minimal vectors in the second exterior power of a lattice, J. Algebra 194 (1997) 467-476. MR 1467163 (98d:11078)

10.
W. Feit, Some lattices over $ {\bf Q}(\sqrt{-3})$, J. Algebra 52 (1978) 248-263. MR 0498407 (58:16534)

11.
Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Math., 1993. MR 1245266 (95c:11044)

12.
M. Kneser, Klassenzahlen definiter quadratischer Formen, Arch. Math. 8 (1957) 241-250. MR 0090606 (19:838c)

13.
J. H. van Lint, F. J. MacWilliams, Generalized quadratic residue codes, IEEE Trans. Inform. Theory 24 (6) (1978) 730-737. MR 514352 (80a:94032)

14.
F. G. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, H. N. Ward, Self-dual codes over $ GF(4)$, J. Combin. Theory Ser. A 25 (1978) 288-318. MR 514624 (80f:94016)

15.
G. Nebe, Finite subgroups of $ GL_n(\mathbb{Q})$ for $ 25\leq n\leq 31$, Comm. Algebra 24 (7) (1996) 2341-2397. MR 1390378 (97e:20066)

16.
W. Plesken, B. Souvignier, Computing isometries of lattices, Computational Algebra and Number Theory (London, 1993), J. Symbolic Comput. 24 (3/4) (1997) 327-334. MR 1484483 (98i:11047)

17.
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 (2) (1995) 190-202. MR 1354045 (96i:11072)

18.
U. Riese, On integral representations for $ {\rm SL}(2,q)$, J. Algebra 242 (2001) 729-739. MR 1848968 (2003c:20013)

19.
R. Scharlau, R. Schulze-Pilot, Extremal lattices, in: B. H. Matzat, G.-M. Greuel, G. Hiss (Eds.), Algorithmic Algebra and Number Theory, Springer, 1998, 139-170. MR 1672117 (99m:11074)

20.
R. Scharlau, A. Schiemann, R. Schulze-Pilot, Theta series of modular, extremal, and hermitian lattices, Proceedings of the Conference on Integral and Quadratic Forms and Lattices, Seoul 1998, Contemporary Math. 249 (1999) 221-233. MR 1732362 (2001a:11066)

21.
A. Schiemann, Classification of hermitian forms with the neighbour method, J. Symbolic Comput. 26 (4) (1998) 487-508. MR 1646667 (99g:11053)

22.
G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954) 274-304. MR 0059914 (15:600b)

23.
H. N. Ward, Quadratic residue codes and symplectic groups, J. Algebra 29 (1974) 150-171. MR 0444290 (56:2648)

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11H06, 11H56, 11E39, 11H71, 11F11

Retrieve articles in all Journals with MSC (2000): 11H06, 11H56, 11E39, 11H71, 11F11

Additional Information:

Kanat Abdukhalikov
Affiliation: Institute of Mathematics, 125 Pushkin Str, 050010, Kazakhstan
Email: abdukhalikov@math.kz

Rudolf Scharlau
Affiliation: Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany
Email: Rudolf.Scharlau@math.uni-dortmund.de

DOI: 10.1090/S0025-5718-08-02131-5
PII: S 0025-5718(08)02131-5
Keywords: Integral lattice, hermitian lattice, extremal lattice, unimodular lattice, root system
Received by editor(s): October 19, 2007
Received by editor(s) in revised form: January 2, 2008
Posted: May 16, 2008
Additional Notes: The first author was supported by the Alexander von Humboldt Foundation.
Copyright of article: Copyright 2008, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google