On Iterative Decoding of Cycle Codes of Graphs

Niclas Wiberg

Iterative decoding techniques have become a viable alternative for constructing high performance coding systems. In particular, the recent success of turbo codes indicates that performance close to the Shannon limit may be achieved. In this thesis, it is showed that many iterative decoding algorithms are special cases of two generic algorithms, the min-sum and sum-product algorithms, which also include non-iterative algorithms such as Viterbi decoding. The min-sum and sum-product algorithms are developed and presented as generalized trellis algorithms, where the time axis of the trellis is replaced by an arbitrary graph, the "Tanner graph". With cycle-free Tanner graphs, the resulting decoding algorithms (e.g., Viterbi decoding) are maximum-likelihood but suffer from an exponentially increasing complexity. Iterative decoding occurs when the Tanner graph has cycles (e.g., turbo codes); the resulting algorithms are in general suboptimal, but significant complexity reductions are possible compared to the cycle-free case. Several performance estimates for iterative decoding are developed, including a generalization of the union bound used with Viterbi decoding and a characterization of errors that are uncorrectable after infinitely many decoding iterations.

Michael Lentmaier, Dmitri V. Truhachev, Kamil Sh. Zigangirov

P. Heijnen, Henk C. A. van Tilborg

Gavin Bernard Horn

In the last six years, we have witnessed an explosion of interest in the coding theory community, in iterative decoding and graphical models, due primarily to the invention of turbo codes. While the structural properties of turbo codes and low density parity check codes have now been put on a firm theoretical footing, what is still lacking is a satisfactory theoretical explanation as to why iterative decoding algorithms perform as well as they do. In this thesis we make a first step by discussing the behavior of various iterative decoders for the graphs of tail-biting codes and cycle codes. By increasing our understanding of the behavior of the iterative min-sum (MSA) and sum-product (SPA) algorithms on graphs with cycles, we can design codes which achieve better performance. Much of this thesis is devoted to the analysis of the performance of the MSA and SPA on the graphs for tail-biting codes and cycle codes. We give sufficient conditions for the MSA to converge to the maximum likelihood codeword after a finite number of iterations. We also use the familiar union bound argument to characterize the performance of the MSA after many iterations. For a cycle code, we show that the performance of the MSA decoder is asymptotically as good as maximum likelihood. For tail-biting codes this will depend on our choice of trellis.

P. Heijnen, F.M.J. Willems, T.J. Tjalkens

Motohiko Isaka, M.P.C. Fossorier, H. Imai

We investigate the property of iterative decoding for finite length codes, and give some insights on the suboptimality of iterative decoding due to the cycles in the graph representation of the codes, for several different types of iteratively decodable codes.