We propose a lattice construction from totally real and complex multiplication fields, which naturally generalizes Construction A of lattices from p-ary codes obtained from the cyclotomic field Q(ζ p ), p a prime, which in turn contains the so-called Construction A of lattices from binary codes as a particular case. We focus on the maximal totally real subfield Q(ζ p r + ζ −r p ) of the cyclotomic field Q(ζ p r ), r ≥ 1. Our construction has applications to coset encoding of algebraic lattice codes for block fading channels, and in particular for block fading wiretap channels.

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D ′ , and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction A ′ of lattices from codes over the polynomial ring F 2 [u]/u a . We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed-Muller codes. In addition, we relate Construction by Code Formula to Construction A ′ by finding a correspondence between nested binary codes and codes over F 2 [u]/u a . This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A ′ . Finally, we show that Construction A ′ produces a lattice if and only if the corresponding code over F 2 [u]/u a is closed under shifted Schur product.

No 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.