Iterative Decoding on Graphs with a Single Cycle’

It is now unders tood [2, 31 t h a t the t u r b o decoding algorithm is a n instance of a probabil i ty propagation a lgor i thm (PPA) o n a g raph wi th many cycles. I n this paper we investigate t h e behavior of a n P P A in graphs wi th a single cycle such as the g raph of a tail-biting code. First , we show t h a t for strictly positive local kernels, t h e i terations of the P P A converge to a unique fixed point, (which was also observed by Anderson a n d Hladik [l] a n d Weiss [ 5 ] ) . Secondly, w e shall generalize a result of McEliece a n d Rodemich [4], by showing that if the hidden variables in t h e cycle are binary-valued, t h e P P A will always make a n opt imal decision. (This was also observed independently by Weiss [ 5 ] ) . W h e n t h e hidden variables can assume 3 or more values, t h e behavior of t h e P P A is much harder to characterize. I. MESSAGE PASSING CONVERGENCE Consider a linear block code described by a tail-biting graph G = (V,E) which consists of a single cycle. We can decode using this graph by passing messages pi , j , between adjacent vertices wi and wj in the cycle. The PPA computes these