On Indexing of Lattice-Constellations for Wireless Network Coding with Modulo-Sum Decoding

We consider wireless (physical-layer) network coding 2-way relaying where both terminals use constellations curved from a common lattice structure and the relay node decodes modulo-sum operation of transmitted data symbols at the multiple-access stage. Performance of such a system with additive white Gaussian noise is strongly determined by used constellation indexing. For some indexing, the minimal distance of modulo-sum decoding is 0 (some points of superimposed- constellation are equal despite corresponding to unequal modulo-sum of data symbols) which causes considerable loss in alphabet-constrained capacity. In this paper, we show that if indices form a modulo-arithmetic progression along each lattice (real) dimension (denoted Affine Indexing (AI)), the minimal distance of modulo-sum decoding equals to minimal distance of primary terminal constellation. Some canonical constellation shapes prevent existence of AI, therefore we propose a greedy-sphere packing algorithm for constellation shape design which jointly maximises minimal distance and keeps existence of AI.