Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Bases in the complex field, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we examine the properties and reduction of such lattices. Focusing on algebraic Lenstra-Lenstra-Lovász (ALLL) reduction, we show that to satisfy Lovás condition requires the ring to be Euclidean. The proposed algorithm can be used to design network coding matrices in compute-and-forward (C & F).