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Belief propagation decoding assisted on-the-fly Gaussian elimination for short LT codes

DOI:10.1007/s10586-015-0522-0
Authors:
Hoyoung Cheong at Namseoul University
Hoyoung Cheong
Jonwon Eun
Jonwon Eun
Jonwon Eun
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Hyuncheol Kim
Hyuncheol Kim
Hyuncheol Kim
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Kuinam J Kim at Kyonggi University
Kuinam J Kim
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Abstract and Figures

Belief propagation (BP) decoding has been widely used for decoding Luby transform (LT) codes which perform very well for a large number of input symbols. However, in reality, small numbers of input symbols are often encountered. In this paper, an efficient BP decoding assisted on-the-fly Gaussian elimination (OFG) decoding process is proposed. Our algorithm exploits XOR operations to get a packet of degree one when the ripple is empty, which gives a small value of overhead. Simulation results show that the proposed algorithm gives a largely improved overhead, about a 0.25 or more, with respect to that of the conventional BP algorithm. The complexity of the proposed algorithm is notably reduced with respect to that of OFG, especially in case of \(k = 150\)–500, while guaranteeing the overhead nearly same as that of OFG.
The number of unused packets of degree 1 after BP decoding successfully (k=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 500, c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
The number of unused packets of degree 1 after BP decoding successfully (k=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 500, c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
… 
The number of operation in OFG decoding with transmitted packets (k=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 500, c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1)
The number of operation in OFG decoding with transmitted packets (k=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 500, c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1)
… 
The number of degree one packets generated from XOR operations
The number of degree one packets generated from XOR operations
… 
The number of operations versus k for BP, OFG and the proposed algorithm (c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1, 0.01 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
The number of operations versus k for BP, OFG and the proposed algorithm (c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1, 0.01 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
… 
The overhead versus k for BP, OFG and the proposed algorithm (c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1, 0.01 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
The overhead versus k for BP, OFG and the proposed algorithm (c=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.1, 0.01 and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0.01)
… 
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Cluster Comput (2016) 19:309–314
DOI 10.1007/s10586-015-0522-0
Belief propagation decoding assisted on-the-fly Gaussian
elimination for short LT codes
Hoyoung Cheong1·Jonwon Eun1·Hyuncheol Kim2·Kuinam J. Kim3
Received: 22 October 2015 / Revised: 14 December 2015 / Accepted: 19 December 2015 / Published online: 1 February 2016
© Springer Science+Business Media New York 2016
Abstract Belief propagation (BP) decoding has been
widely used for decoding Luby transform (LT) codes which
perform very well for a large number of input symbols. How-
ever, in reality, small numbers of input symbols are often
encountered. In this paper, an efficient BP decoding assisted
on-the-fly Gaussian elimination (OFG) decoding process is
proposed. Our algorithm exploits XOR operations to get a
packet of degree one when the ripple is empty, which gives a
small value of overhead. Simulation results show that the pro-
posed algorithm gives a largely improved overhead, about a
0.25 or more, with respect to that of the conventional BP algo-
rithm. The complexity of the proposed algorithm is notably
reduced with respect to that of OFG, especially in case of
k=150–500, while guaranteeing the overhead nearly same
as that of OFG.
Keywords OFG decoding ·Complexity ·
Triangularization ·Gaussian elimination
BKuinam J. Kim
kuinamj@gmail.com
Hoyoung Cheong
hycheong@nsu.ac.kr
Jonwon Eun
jweun@nsu.ac.kr
Hyuncheol Kim
hckim@nsu.ac.kr
1Department of Information Communication, Namseoul
University, Cheonan, Korea
2Department of Computer Science, Namseoul University,
Cheonan 331-707, Korea
3Department of Convergence Security, Kyonggi University,
Suwon, Korea
1 Introduction
Fountain codes are a promising solution to multicast reli-
able information with a low complexity over binary erasure
channel (BEC). Luby transform (LT) codes, which are the
first implementation of Fountain codes, approaches capacity
for increasing code length, k[1]. However, in reality, short
length messages are often encountered. In video streaming
applications, the number of frames in a group of pictures can
be a message length, which is typically small value. Under
this configuration of a small or limited number of symbols
for coding in LT codes, mathematical analysis and simulation
results have shown that LT codes can perform poorly [2,3].
There are several decoding algorithms for LT codes and the
widely used is belief propagation (BP) decoding algorithm.
The complexity of BP is very low and its decoding speed is
fast, but for a small value of k, it requires a large overhead.
GE or GE-like algorithm, such as On-the-fly Gaussian elim-
ination (OFG) or incremental Gaussian elimination (IG) [4],
shows a noticeable overhead performance for the small value
of k[5].
In the binary erasure channel, the BP decoding process
can be significantly simplified since all received symbols
in decoding are completely reliable. BP first finds degree-1
received symbols and moves it into theripple. Symbols in the
ripple are processed one by one to decode another symbols
with degree more than one. The ripple in the BP decoding
process plays an important role for decoding LT codes suc-
cessfully. When there is no degree-1 packets in the ripple,
BP decoder declares the decoding failure. In case of decod-
ing failure a new packet is received, the 1sin the decoded
positions of the corresponding equation are canceled and BP
decoding is reattempted. Thats why the BP decoder has a
large overhead for a small value of k[5]. BP decoding has
been widely used for decoding LT codes which perform very
123
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... In the case of small numbers of input symbols, just like our research case, the second type of treatment for decoding LT codes over an AWGN channel were proposed. Cheong et al [12] utilised a BP decoding algorithm assisted on-the-fly Gaussian elimination (OFG). The study tries to overcome the problem of missing degree one coded symbols by using xoring operations for certain coded symbols to reproduce a degree one coded symbol again. ...
... It is obvious that, when dealing with LT codes in an AWGN channel, the prementioned studies [9][10][11][12][13][14] try to overcome the problems of error propagation by adapting the decoding matrix of the LT code so that it is fitted with that of LDPC codes in order to use efficient iterative soft decoding. But this efficient decoding entails large complexity due to its iterative nature. ...
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