Conditionally Cycle-Free Graphical Models for Coset Codes

Conditionally cycle-free graphical models (i.e., cyclic graphical models which become cycle-free after conditioning on a subset of the hidden variables) are constructed for coset codes. Following the description of a general construction procedure, examples of a number of families of codes-including first-order Reed-Muller (RM) and the Delsarte-Goethals codes - are provided for which the proposed procedure yields optimal soft-in soft-out (SISO) decoding algorithms that are less complex than the best known trellis-based algorithms. In the case of the first-order RM codes, which have a recursive coset construction, the optimal SISO decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation (RCR) decoding. Connections are made between RCR decoding and existing algorithms that exploit fast Hadamard transforms. Finally, the utility of the proposed decoding algorithms are supported by a practically motivated application: the construction of serially concatenated codes that have high rates and low error floors. Extended Hamming codes are proposed as outer codes in such constructions with efficient decoding employing the SISO algorithms developed herein.