Constructions a of lattices from number fields and division algebras

Abstract

There is a rich theory of relations between lattices and linear codes over finite fields. However, this theory has been developed mostly with lattice codes for the Gaussian channel in mind. In particular, different versions of what is called Construction A have connected the Hamming distance of the linear code to the Euclidean structure of the lattice.This paper concentrates on developing a similar theory, but for fading channel coding instead. First, two versions of Construction A from number fields are given. These are then extended to division algebra lattices. Instead of the Euclidean distance, the Hamming distance of the finite codes is connected to the product distance of the resulting lattices, that is the minimum product distance and the minimum determinant respectively.

Algebraic lattices achieving the capacity of the ergodic fading channel

“…It was proven in [6], for iid fading processes under some regularity conditions, that there exists a sequence of fadinggood lattices with P e (Λ) = O(1/n 2 ). The proof only requires an MH ensemble, and hence an immediate corollary of Theorem 1 is that Generalized Construction A lattices as in (10) are also fading good. We provide next a different approach that explores the algebraic structure and obtains an exponential decay of P e (Λ) with respect to n in iid fading channels.…”

“…We now describe the algebraic Construction A proposed in [10]. We recall some main definitions and results, but assume some basic familiarity with Algebraic Number Theory.…”

“…The properties of Λ K (C) are studied in [10]. First of all Λ K (C) is a full rank lattice and Λ K ({0}) = φ(pO K ) is a sublattice with index |C| = p k .…”

“…• A Minkowski-Hlawka theorem for a generalized (algebraic) version of Construction A lattices (Theorem 1). This Construction A was first proposed in [10], but its asymptotic goodness was up to now unknown. • Properties of the group of units in the ring of integers (lemmas 1 and 2).…”

Universal Lattice Codes for MIMO Channels

“…The Random Ensemble 1) Construction A: Here we present an algebraic Construction A suitable for the ergodic fading model. This construction differs slightly from the one in V-A2 and was firstly studied in [25].…”