Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T17:06:35.241Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on lattices of rational functions

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

H. -G. Quebbemann
H. -G. Quebbemann
Affiliation:
Professor H.-G. Quebbemann, FB6-Mathematik, Universität Oldenburg, Ammerländer Heerstrasse 114-118, D-2900 Oldenburg, Deutschland.
Get access

Extract

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field, and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).

MSC classification

Secondary: 11H31: Lattice packing and covering
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 10 - 13
Copyright
Copyright © University College London 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Conway, J. H. and Sloane, N. J. A.. Sphere Packings, Lattices and Groups (Springer-Verlag, 1988).CrossRefGoogle Scholar
2. Feit, W.. On integral representations of finite groups. Proc. London Math. Soc. (3), 29 (1974), 633–683.CrossRefGoogle Scholar
3. Lint, J. H. van. Introduction to Coding Theory (Springer-Verlag, 1982).CrossRefGoogle Scholar
4. Quebbemann, H. G.. Estimates of regulators and class numbers in function fields. To appear in J. Reine u. Anger. Math.Google Scholar
5. Rosenbloom, M. Y. and Tsfasman, M. A.. Multiplicative lattices in global fields. Invent, math., 101 (1990), 687–696.CrossRefGoogle Scholar
6. Rush, J. A.. A lower bound on packing density. Invent, math., 98 (1989), 499–509.CrossRefGoogle Scholar