Lattice Codes for Physical Layer Communications

Lattices are deceptively simple mathematical structures that have become indispensable for code design for physical layer communications. While lattice-related problems are interesting in their own right, the usefulness of these discrete structures in wireless communications provides additional motivation for their study and enables a multidisciplinary line of research. This thesis is devoted to the study of lattice code design for physical layer communications. Modern wireless communication networks are required to accommodate significantly varied types of mobile devices, differing in available computational power or number of equipped antennas. Additionally, the density of the networks increases rapidly, and many communication protocols diverge from the classical direct point-to-point transmission in favor of allowing for intermediate relays to process and forward data. An important consequence of this shift towards more sophisticated transmission protocols is that traditional well-performing codes become futile for modern communications, thus the study and development of novel codes is called for. Yet, however involved a transmission protocol may be, the characteristics of the physical medium, i.e., the wireless channel, stay unaffected. It is thus natural that an underlying lattice structure for code design remains crucial. This thesis consists of several articles considering lattice code design for four different communication settings relevant in modern wireless communications. We begin by studying two communication scenarios for which space-time lattice codes, objects studied since 1998, arise naturally due to the characteristics of the transmission protocol. The first considered setup is an asymmetric point-to-point channel, for which we construct full-rank matrix lattices which serve as the underlying structure for code construction. In particular, we are interested in an invariant of certain orders in the considered algebra, called discriminant. We then move on to a relaying technique known as amplify-and-forward. We propose constructions of well-performing space-time lattice codes adopted to this particular setting, which additionally allow for a significant reduction in decoding complexity. The other two scenarios considered make use of Voronoi codes, also referred to as nested lattice codes, a concept first introduced in 1983. We study maximum-likelihood decoding in the context of a relaying protocol known as compute-and-forward, and are furthermore concerned with the message recoverability at the destination. Finally, we consider the wiretap channel and are particularly interested in the performance with respect to a potential eavesdropper. In the latter two scenarios, we derive design criteria related to the theta series of certain lattices involved in the code design.