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Number Theory

Geometry of Numbers

11Hxx

[1] Kanat Abdukhalikov. Unimodular Hermitian lattices. Mathematisches Forschungsinstitut Oberwolfach Report No. 1/2005, pages 27–30, 2005.
[2] Kanat Abdukhalikov and Rudolf Scharlau. Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers. Math. Comp., 78(265):387–403, 2009.
[3] Ali Akhavi and Damien Stehlé. Speeding-up lattice reduction with random projections (extended abstract). In LATIN 2008: Theoretical Informatics, volume 4957 of Lecture Notes in Computer Science, pages 293–305. Springer, 2008.
[4] Christine Bachoc and Gabriele Nebe. Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order. Experiment. Math., 6(2):151–162, 1997.
[5] Christine Bachoc and Boris Venkov. Modular forms, lattices and spherical designs. In Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, volume 37 of Monogr. Enseign. Math., pages 87–111. Enseignement Math., Geneva, 2001.
[6] Werner Backes and Susanne Wetzel. Heuristics on lattice basis reduction in practice. ACM J. Exp. Algorithmics, 7:21 pp. (electronic), 2002.
[7] Robin Chapman, Steven T. Dougherty, Philippe Gaborit, and Patrick Solé. 2-modular lattices from ternary codes. J. Théor. Nombres Bordeaux, 14(1):73–85, 2002.
[8] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, volume 290 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, third edition, 1999.
[9] Mathieu Dutour Sikirić, Achill Schürmann, and Frank Vallentin. A generalization of Voronoi's reduction theory and its application. Duke Math. J., 142(1):127–164, 2008.
[10] C. Fieker and M. E. Pohst. On lattices over number fields. In Algorithmic Number Theory (Talence, 1996), volume 1122 of Lecture Notes in Comput. Sci., pages 133–139. Springer, Berlin, 1996.
[11] Philippe Gaborit. Construction of new extremal unimodular lattices. European J. Combin., 25(4):549–564, 2004.
[12] Masaaki Harada. On the existence of frames of the Niemeier lattices and self-dual codes over Fp. J. Algebra, 321(8):2345–2352, 2009.
[13] Masaaki Harada, Masaaki Kitazume, and Michio Ozeki. Ternary code construction of unimodular lattices and self-dual codes over Z6. J. Algebraic Combin., 16(2):209–223, 2002.
[14] Boris Hemkemeier. Algorithmische konstruktionen von gittern. arXiv:math.MG/0411134, 64 pages, 2004.
[15] G. Nebe. Kneser-Hecke-operators in coding theory. Abh. Math. Sem. Univ. Hamburg, 76:79–90, 2006.
[16] Gabriele Nebe. Finite quaternionic matrix groups. Represent. Theory, 2:106–223 (electronic), 1998.
[17] Gabriele Nebe. Even lattices with covering radius <√2. Beiträge Algebra Geom., 44(1):229–234, 2003.
[18] Gabriele Nebe. Strongly modular lattices with long shadow. J. Théor. Nombres Bordeaux, 16(1):187–196, 2004.
[19] Gabriele Nebe and Kristina Schindelar. S-extremal strongly modular lattices. J. Théor. Nombres Bordeaux, 19(3):683–701, 2007.
[20] Gabriele Nebe and Boris Venkov. The strongly perfect lattices of dimension 10. J. Théor. Nombres Bordeaux, 12(2):503–518, 2000.
[21] Gabriele Nebe and Boris Venkov. Low-dimensional strongly perfect lattices I: The 12-dimensional case. Enseign. Math. (2), 51(1-2):129–163, 2005.
[22] Gabriele Nebe and Boris Venkov. Low dimensional strongly perfect lattices III: Dual strongly perfect lattices of dimension 14. arXiv:0809.0593v1 [math.NT], 33 pages, 2008.
[23] Gabriele Nebe and Chaoping Xing. A Gilbert-Varshamov type bound for Euclidean packings. Math. Comp., 77(264):2339–2344, 2008.
[24] Phong Q. Nguên and Damien Stehlé. Floating-point LLL revisited. In Advances in Cryptology - Eurocrypt 2005, Lecture Notes in Computer Science, pages 215–233. Springer Berlin/Heidelberg, 2005.
[25] Phong Q. Nguyen and Damien Stehlé. LLL on the average. In Algorithmic Number Theory, volume 4076 of Lecture Notes in Comput. Sci., pages 238–256. Springer, Berlin, 2006.
[26] W. Plesken and M. Pohst. Constructing integral lattices with prescribed minimum. I. Math. Comp., 45(171):209–221, S5–S16, 1985.
[27] E. M. Rains and N. J. A. Sloane. The shadow theory of modular and unimodular lattices. J. Number Theory, 73(2):359–389, 1998.
[28] Achill Schuermann. Perfect, strongly eutactic lattices are periodic extreme. arXiv:0808.2013v1 [math.MG], 18 pages, 2008.
[29] Achill Schürmann and Frank Vallentin. Local covering optimality of lattices: Leech lattice versus root lattice E8. Int. Math. Res. Not., (32):1937–1955, 2005.
[30] Mathieu Dutour Sikirić, Achill Schürmann, and Frank Vallentin. Classification of eight-dimensional perfect forms. Electron. Res. Announc. Amer. Math. Soc., 13:21–32 (electronic), 2007.
[31] N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway. Minimal-energy clusters of hard spheres. Discrete Comput. Geom., 14(3):237–259, 1995.
[32] Dan Yasaki. Binary Hermitian forms over a cyclotomic field. J. Algebra, In Press, 2009.

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