On the densest MIMO lattices from cyclic division algebras

TL;DR

This paper investigates how to construct the densest MIMO lattice codes using cyclic division algebras by minimizing discriminants, providing theoretical bounds, explicit examples, and improved algorithms.

Contribution

It establishes a lower bound on the discriminant of maximal orders in cyclic division algebras and presents methods to achieve this bound for optimal lattice density.

Findings

Derived a lower bound for the minimal discriminant using class field theory.

Constructed a lattice with 2.5 times more codewords than the Golden code at the same minimum determinant.

Enhanced existing algorithms for finding maximal orders in cyclic division algebras.

Abstract

It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving our bound are given. E.g. we construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We describe a general algorithm due to Ivanyos and Ronyai for finding maximal orders within a cyclic division algebra and discuss our enhancements to this algorithm. We also consider general methods for finding cyclic division algebras of a prescribed index achieving our lower bound.