On Integral Well-Rounded Lattices in the Plane
Document Type
Article
Department
Mathematics (CMC)
Publication Date
10-2012
Abstract
We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form x 2+ Dy 2= z 2 where D>0 is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.
Rights Information
© 2012 Springer Science+Business Media, LLC
DOI
10.1007/s00454-012-9432-6
Recommended Citation
Fukshansky, Lenny, Glenn Henshaw, Philip Liao, Matthew Prince, Xun Sun, and Samuel Whitehead. "On Integral Well-Rounded Lattices in the Plane." Discrete and Computational Geometry 48.3 (2012): 735-748. doi: 10.1007/s00454-012-9432-6
Comments
Please note that this article was originally published by Springer-Verlag in Discrete & Computational Geometry and therefore the final publication is available at http://link.springer.com/article/10.1007/s00454-012-9432-6.