Hierarchical Pairwise Error Probability for Hierarchical Decode and Forward Strategy in PLNC

Pairwise error probability (PEP) is an essential tool allowing to design practical optimized coding schemes. It reveals the connection between the performance and the decoding metric that is directly related to the codeword and/or constellation properties. The application of this principle to the
hierarchical
PEP used to describe the decoding performance of
many-to-one
message functions (hierarchical network code maps) in physical layer network coding (PLNC) is addressed in this letter. Unlike for the single-user case, the
hierarchical PEP
reveals a complicated dependence on the structure of the hierarchical codeword/constellation. The structure is defined in terms of
hierarchical distance
and
hierarchical self-distance spectra
. We show that the network coded modulation (NCM) minimizing the hierarchical decoding error probability should have zero self-distance spectrum leading to
self-folded NCM

design criterion
.