Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function

We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call the symbolwise graph-cover decoder, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help towards a better understanding of message-passing iterative decoding, namely blockwise graphcover decoding links max-product (min-sum) algorithm decoding with li...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Pascal O. Vontobel
• Keywords: Bethe Entropy Function, Combinatorial Characterization, CORR 2010, Education, Factor Graph