On integral well-rounded lattices in the plane

TL;DR

This paper classifies integral well-rounded lattices in the plane using Pell-type equations, analyzes their properties such as minimal norm and signal-to-noise ratio, and explores applications in lattice-based communication networks.

Contribution

It provides a new parameterization of these lattices via Pell equations and extends previous classifications to broader cases.

Findings

Parameterization of lattices via Pell equations.

Estimation of lattice properties like minimal norm and signal-to-noise ratio.

Analysis of the number of such lattices for each determinant.

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.