Well-Rounded Twists of Ideal Lattices from Real Quadratic Fields

TL;DR

This paper investigates ideal lattices from real quadratic fields, providing methods to compute well-rounded twists, and constructs infinite families of lattices with optimal packing and distance properties, including a classification of certain fields.

Contribution

It introduces an explicit method for finding all well-rounded twists of ideal lattices from real quadratic fields and classifies fields with unique well-rounded twists.

Findings

Constructed infinite families of lattices with good sphere-packing radius.

Identified conditions for the orthogonal lattice to be the only well-rounded twist.

Provided a complete classification of relevant real quadratic fields.

Abstract

We study ideal lattices in $\mathbb{R}^2$ coming from real quadratic fields, and give an explicit method for computing all well-rounded twists of any such ideal lattice. We apply this to ideal lattices coming from Markoff numbers to construct infinite families of non-equivalent planar lattices with good sphere-packing radius and good minimum product distance. We also provide a complete classification of all real quadratic fields such that the orthogonal lattice is the only well-rounded twist of the lattice corresponding to the ring of integers.