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arXiv:cs/0611097 · IEEE International Symposium on Information Theory (ISIT)

Conditionally Cycle-Free Generalized Tanner Graphs: Theory and Application to High-Rate Serially Concatenated Codes

Thomas R. Halford, K.M. Chugg

Published Jun 1, 2007• 3 citations
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Abstract

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.

DOI
10.1109/isit.2007.4557495

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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.
Generalized Tannergraphshavebeenimplicitly studied byanumberofauthors undertherubric ofgeneralized parity-check matrices. Thisworkconsiders theconditioning of binary hiddenvariables insuchmodelsinordertobreakall cycles andthusderive optimal soft-in soft-out (SISO) decoding algorithms. Conditionally cycle-free generalized Tannergraphs areshowntoimply optimal SISOdecoding algorithms forthefirst order Reed-Muller codes andtheir duals -theextended Hamming codes-whicharesubstantially lesscomplex thanconventional bit-level trellis decoding. Thestudyoflow-complexity optimal SISOdecoding algorithms forthefamily ofextended Hamming codesispractically motivated. Specifically, itisshownthat exended Hammingcodesoffer anattractive alternative tohigh- rateconvolutional codesintermsofbothperformance and complexity foruseinveryhigh-rate, verylow-floor, serially concatenated coding schemes.
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.
In this work, we consider iterative soft-decision decoding of binary linear codes using a generalized Tanner graph. A generalized Tanner graph is constructed from the code's Tanner graph representation under the objective of mitigating the effect of small cycles. Then, iterative decoding is applied to the generalized Tanner graph using trellis-based soft-input soft-output decoding in the generalized check nodes. When combined with the stochastic shifting algorithm of Jiang and Narayanan (IEEE Commun. Letters, 2004), the proposed decoding method provides both improved error rate performance and reduced average decoding complexity, without increasing the worst-case decoding complexity.
A number of authors have recently considered iterative soft-in soft-out (SISO) decoding algorithms for classical linear block codes that utilize redundant Tanner graphs. Jiang and Narayanan presented a practically realizable algorithm that applies only to cyclic codes while Kothiyal et al. presented an algorithm that, while applicable to arbitrary linear block codes, does not imply a low-complexity implementation. This work first presents the aforementioned algorithms in a common framework and then presents a related algorithm - random redundant iterative decoding - that is both practically realizable and applicable to arbitrary linear block codes. Simulation results illustrate the successful application of the random redundant iterative decoding algorithm to the extended binary Golay code. Additionally, the proposed algorithm is shown to outperform Jiang and Narayanan's algorithm for a number of Bose-Chaudhuri-Hocquenghem (BCH) codes
Constructs Reed-Muller codes by generalized multiple concatenation of binary block codes of length 2. As a consequence of this construction, a new decoding procedure is derived that uses soft-decision information. The algorithm is designed for low decoding complexity and is applicable to all Reed-Muller codes. It gives better decoding performance than soft-decision bounded-distance decoding. Its decoding complexity is much lower than that of maximum-likelihood trellis decoding of Reed-Muller codes, especially for long codes.>
We study the design of structured Tanner codes with low error-rate floors on the AWGN channel. The design technique involves the “doping” of standard LDPC (proto-)graphs, by which we mean Hamming or recursive systematic convolutional (RSC) code constraints are used together with single-parity-check (SPC) constraints to construct a code’s protograph. We show that the doping of a “good” graph with Hamming or RSC codes is a pragmatic approach that frequently results in a code with a good threshold and very low error-rate floor. We focus on low-rate Tanner codes, in part because the design of low-rate, low-floor LDPC codes is particularly difficult. Lastly, we perform a simple complexity analysis of our Tanner codes and examine the performance of lower-complexity, suboptimal Hamming-node decoders.