Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

Abdukhalikov, Kanat; Scharlau, Rudolf

doi:10.1090/S0025-5718-08-02131-5

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers
HTML articles powered by AMS MathViewer

by Kanat Abdukhalikov and Rudolf Scharlau;

Math. Comp. 78 (2009), 387-403

DOI: https://doi.org/10.1090/S0025-5718-08-02131-5

Published electronically: May 16, 2008

PDF | Request permission

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

K. S. Abdukhalikov, Invariant Hermitian lattices in the Steinberg module and their isometry groups, Comm. Algebra 25 (1997), no. 8, 2607–2626. MR 1459581, DOI 10.1080/00927879708826010
Kanat Abdukhalikov, Unimodular Hermitian lattices in dimension 13, J. Algebra 272 (2004), no. 1, 186–190. MR 2029031, DOI 10.1016/j.jalgebra.2003.05.006
Christine Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A 78 (1997), no. 1, 92–119. MR 1439633, DOI 10.1006/jcta.1996.2763
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436. MR 422448
John H. Conway, Vera Pless, and Neil J. A. Sloane, Self-dual codes over $\textrm {GF}(3)$ and $\textrm {GF}(4)$ of length not exceeding $16$, IEEE Trans. Inform. Theory 25 (1979), no. 3, 312–322. MR 528009, DOI 10.1109/TIT.1979.1056047
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
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
Renaud Coulangeon, Minimal vectors in the second exterior power of a lattice, J. Algebra 194 (1997), no. 2, 467–476. MR 1467163, DOI 10.1006/jabr.1997.7039
Walter Feit, Some lattices over $\textbf {Q}(\surd -3)$, J. Algebra 52 (1978), no. 1, 248–263. MR 498407, DOI 10.1016/0021-8693(78)90273-9
Yoshiyuki Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106, Cambridge University Press, Cambridge, 1993. MR 1245266, DOI 10.1017/CBO9780511666155
Martin Kneser, Klassenzahlen definiter quadratischer Formen, Arch. Math. 8 (1957), 241–250 (German). MR 90606, DOI 10.1007/BF01898782
Jacobus H. van Lint and F. Jessie MacWilliams, Generalized quadratic residue codes, IEEE Trans. Inform. Theory 24 (1978), no. 6, 730–737. MR 514352, DOI 10.1109/TIT.1978.1055965
F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, Self-dual codes over $\textrm {GF}(4)$, J. Combin. Theory Ser. A 25 (1978), no. 3, 288–318. MR 514624, DOI 10.1016/0097-3165(78)90021-3
Gabriele Nebe, Finite subgroups of $\textrm {GL}_n(\mathbf Q)$ for $25\leq n\leq 31$, Comm. Algebra 24 (1996), no. 7, 2341–2397. MR 1390378, DOI 10.1080/00927879608825704
W. Plesken and B. Souvignier, Computing isometries of lattices, J. Symbolic Comput. 24 (1997), no. 3-4, 327–334. Computational algebra and number theory (London, 1993). MR 1484483, DOI 10.1006/jsco.1996.0130
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 (1995), no. 2, 190–202. MR 1354045, DOI 10.1006/jnth.1995.1111
Udo Riese, On integral representations for $\textrm {SL}(2,q)$, J. Algebra 242 (2001), no. 2, 729–739. MR 1848968, DOI 10.1006/jabr.2001.8792
Rudolf Scharlau and Rainer Schulze-Pillot, Extremal lattices, Algorithmic algebra and number theory (Heidelberg, 1997) Springer, Berlin, 1999, pp. 139–170. MR 1672117
Rudolf Scharlau, Alexander Schiemann, and Rainer Schulze-Pillot, Theta series of modular, extremal, and Hermitian lattices, Integral quadratic forms and lattices (Seoul, 1998) Contemp. Math., vol. 249, Amer. Math. Soc., Providence, RI, 1999, pp. 221–233. MR 1732362, DOI 10.1090/conm/249/03760
Alexander Schiemann, Classification of Hermitian forms with the neighbour method, J. Symbolic Comput. 26 (1998), no. 4, 487–508. MR 1646667, DOI 10.1006/jsco.1998.0225
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Harold N. Ward, Quadratic residue codes and symplectic groups, J. Algebra 29 (1974), 150–171. MR 444290, DOI 10.1016/0021-8693(74)90120-3