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The IMA volumes in mathematics and its applications

An Algebraic Description of Iterative Decoding Schemes

Elke Offer, Emina Soljanin

Published Jan 1, 2001• 4 citations
Publisher
DOI
10.1007/978-1-4613-0165-3_15

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In this paper, we describe two novel modifications of an algorithm for deriving sparse parity-check matrix representations of classical binary and non-binary algebraic codes. The proposed sparse representations are obtained by introducing a small number of auxiliary variables and check-equations that lead to Tanner graphs without four-cycles. When used with iterative decoding, the developed code representations show significant gain in performance when compared to other known description techniques.
Bit-by-bit soft-decision decoding of binary cyclic codes is considered. A significant reduction in decoder complexity can be achieved by requiring only that the decoder correct all analog error patterns which fall within a Euclidean sphere whose radius is equal to half the minimum Euclidean distance of the code. Such a "maximum-radius" scheme is asymptotically optimum for the additive white Gaussian noise (AWGN) channel. An iterative extension of the basic algebraic analog decoding scheme is discussed, and performance curves are given for the (17,9), (21,11), and (73,45) codes on the AWGN channel.
We investigate iterative algebraic decoding of codes defined on graphs. Practical codes that can achieve the Shannon limit are codes defined on graphs with its associated iterative decoding algorithms. Yet, the complexity of these algorithms is high, especially for long codes that attain capacity. For the foreseeable future, algebraic decoding will still be a standard in industry. We aim to harness the power of iterative processes and the low complexity of algebraic techniques by decoding codes on graphs using iterative algebraic techniques. We study the performance of codes on graphs decoded with this method taking into account the case of undetected errors. More specifically, we examine the threshold under which decoding is successful for the BEC and the BSC and compare these with that of certain LDPC codes. Our initial investigation shows that although a significant loss of performance is incurred on the BEC when compared to belief propagation decoding, for transmission over the BSC the threshold value is somewhat close to belief propagation. Especially for high rate codes iterative algebraic decoding could have good performance while maintaining low decoding complexity.