Optimal Two-Dimensional Lattices for Precoding of Linear Channels

Research output: Contribution to journalArticle


Consider the communication system model y = HFx+n, where H and F are the channel and precoder matrices, x is a vector of data symbols drawn from some lattice-type constellation, such as M-QAM, n is an additive white Gaussian noise vector and y is the received vector. It is assumed that both the transmitter and the receiver have perfect knowledge of the channel matrix H and that the transmitted signal Fx is subject to an average energy constraint. The columns of the matrix HF can be viewed as the basis vectors that span a lattice, and we are interested in the precoder F that maximizes the minimum distance of this lattice. This particular problem remains open within the theory of lattices and the communication theory. This paper provides the complete solution for any non-singular M x 2 channel matrix H. For real-valued matrices and vectors, the solution is that HF spans the hexagonal lattice. For complex-valued matrices and vectors, the solution is that HF, when viewed in four-dimensional real-valued space, spans the Schlafli lattice D-4.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Electrical Engineering, Electronic Engineering, Information Engineering


  • Two-dimensional lattices, precoding, linear channel
Original languageEnglish
Pages (from-to)2104-2113
JournalIEEE Transactions on Wireless Communications
Issue number5
Publication statusPublished - 2013
Publication categoryResearch