Lattices from elliptic curves over finite fields. (English) Zbl 1296.11086
Summary: In their well known book M. A. Tsfasman and S. G. Vlǎduţ [Algebraic-geometric codes. Transl. from the Russian. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0727.94007)] introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.
MSC:
| 11H06 | Lattices and convex bodies (number-theoretic aspects) |
| 11G20 | Curves over finite and local fields |
Citations:
Zbl 0727.94007References:
| [1] | Cohn, H.; Kumar, A., Optimality and uniqueness of the Leech lattice among lattices, Ann. Math. (2), 170, 3, 1003-1050 (2009) · Zbl 1213.11144 |
| [2] | Conway, J. H.; Sloane, N. J.A., Sphere Packings, Lattices, and Groups (1999), Springer-Verlag · Zbl 0915.52003 |
| [3] | Gruber, P. M.; Lekkerkerker, C. G., Geometry of Numbers (1987), North-Holland Publishing Co. · Zbl 0611.10017 |
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| [5] | Stichtenoth, H., Algebraic Function Fields and Codes (2009), Springer: Springer Berlin · Zbl 1155.14022 |
| [6] | Tsfasman, M. A.; Vladut, S. G., Algebraic-Geometric Codes (1991), Kluwer Academic Publishers · Zbl 0727.94007 |
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