Enabling multiplication in lattice codes via Construction A

As a first step towards distributed computations in a wireless network, we introduce ideal lattices, that is lattices built over an ideal of a ring of integers in a number field, as a tool for constructing lattice codes at the physical layer. These lattices are not only additive groups as all lattices, but they are also equipped with a multiplication, which enables polynomial operations at each node of the wireless network. In this paper, we show how some of these ideal lattices can be constructed from polynomial codes (generalization of cyclic codes) via Construction A, and illustrate how these lattices enable multiplication.