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Hierarchical-Decode-and-Forward is a promising wireless-network-coding-based 2-way relaying strategy due to its potential to operate outside the classical multiple-access capacity region. Assuming a practical scenario with channel state information at the receiver and no channel adaptation, there exist modulations and exclusive codes for which even non-zero channel parameters (denoted as catastrophic) cause zero hierarchical minimal distance -- significantly degrading its performance. In this work, we state that non-binary linear alphabets cannot avoid these parameters and some exclusive codes even imply them; contrary XOR does not. We define alphabets avoiding all catastrophic parameters and reaching its upper bound on minimal distance for all parameter values (denoted uniformly most powerful (UMP)). We find that binary, non-binary orthogonal and bi-orthogonal modulations are UMP. We optimize scalar parameters of FSK (modulation index) and full-response CPM (frequency pulse shape) to yield UMP alphabets.

Design of Uniformly Most Powerful Alphabets for HDF 2-Way Relaying Employing Non-Linear Frequency Modulations