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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

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