Design criteria for lattice network coding

The compute-and-forward (C-F) relaying strategy proposed by Nazer and Gastpar is a powerful new approach to physical-layer network coding. Nazer-Gastpars construction of C-F codes relies on asymptotically-good lattice partitions that require the dimension of lattices to tend to infinity. Yet it remains unclear how such C-F codes can be constructed and analyzed under practical constraints. Motivated by this, an algebraic approach was taken to compute-and-forward, which provides a framework to study C-F codes constructed from finite-dimensional lattice partitions. Building on the algebraic framework, this paper moves one step further; it aims to derive the design criteria for the C-F codes constructed from finite-dimensional lattice partitions (also referred to as lattice network codes). It is shown that the receiver parameters {a_ℓ} and α should be chosen such that the quantity Q = |α|² + SNRΣ_ℓ=1^L ||αh_ℓ - αℓ||² is minimized, and the lattice partition should be designed such that the minimum inter-coset distance is maximized. These design criteria imply that finding the optimal receiver parameters is equivalent to solving a shortest vector problem, and designing good lattice partitions can be reduced to the design of good linear codes for complex Construction A.