Arithmetic geometry of compute and forward

We propose the joint study of function computation (arithmetics) and lattice coding gain (geometry) to derive the (complex modulo) arithmetics of compute and forward over Euclidean geometry. First, we demonstrate that only five families of complex alphabets exist that admit euclidean complex modulo arithmetics. Second, we prove that the (per-dimension) euclidean division algorithm is equivalent to a closest vector algorithm hence a natural framework for compute and forward. Third, we derive the nominal coding gains of the five resulting families of (nested) lattice codes obtained as preimages of linear block codes. Finally we apply the proposed arithmetic geometry framework to the MAC channel and show graphical illustration of the 2-user case over the Eisenstein numbers.