  Generalized Tanner graphs have been implicitly studied by a number of authors
under the rubric of generalized parity-check matrices. This work considers the
conditioning of binary hidden variables in such models in order to break all
cycles and thus derive optimal soft-in soft-out (SISO) decoding algorithms.
Conditionally cycle-free generalized Tanner graphs are shown to imply optimal
SISO decoding algorithms for the first order Reed-Muller codes and their duals
- the extended Hamming codes - which are substantially less complex than
conventional bit-level trellis decoding. The study of low-complexity optimal
SISO decoding algorithms for the family of extended Hamming codes is
practically motivated. Specifically, it is shown that exended Hamming codes
offer an attractive alternative to high-rate convolutional codes in terms of
both performance and complexity for use in very high-rate, very low-floor,
serially concatenated coding schemes.
