Triangular network coding - Revision history

2001:56A:E8AA:600:E4AB:12C4:57BF:1317 at 22:18, 14 June 2024

2024-06-14T22:18:36Z

← Previous revision		Revision as of 22:18, 14 June 2024
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	The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869–878 \|doi=10.1016/j.dcan.2022.12.006 \|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \| last2 = Foh \| first2 = Chuan Heng \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869–878 \|doi=10.1016/j.dcan.2022.12.006 \|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>

	==Coding and decoding==		==Coding and decoding==

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2024-05-15T05:48:42Z

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← Previous revision		Revision as of 05:48, 15 May 2024
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	The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869–878 \|doi=10.1016/j.dcan.2022.12.006 ~~\|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747~~\|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869–878 \|doi=10.1016/j.dcan.2022.12.006 \|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>

	==Coding and decoding==		==Coding and decoding==

Mazewaxie: WP:GENFIXES

2024-03-07T21:15:42Z

WP:GENFIXES

← Previous revision		Revision as of 21:15, 7 March 2024
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	The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=~~869-878~~ \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747\|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869–878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747\|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>

	==Coding and decoding==		==Coding and decoding==
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	In TNC, coding is performed in two stages. First redundant "0" bits are added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are [[Exclusive or\|XOR coded]], bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a [[Triangular matrix\|triangular pattern]].		In TNC, coding is performed in two stages. First redundant "0" bits are added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are [[Exclusive or\|XOR coded]], bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a [[Triangular matrix\|triangular pattern]].

	In essence, the TNC decoding process, like the LNC decoding process involves [[Gaussian elimination]]. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of ''triangularization,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with complexity of <math>O(n^3)</math>, where <math>n</math> is the number of packets, can be bypassed. The receiver now only needs to perform ''back-substitution,''<ref name=fraleigh95~~>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.<~~/~~ref~~> with worst-case complexity given as <math>O(n^2)</math> for each bit location.		In essence, the TNC decoding process, like the LNC decoding process involves [[Gaussian elimination]]. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of ''triangularization,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with complexity of <math>O(n^3)</math>, where <math>n</math> is the number of packets, can be bypassed. The receiver now only needs to perform ''back-substitution,''<ref name=fraleigh95/> with worst-case complexity given as <math>O(n^2)</math> for each bit location.

	==References==		==References==
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	[[Category:Finite fields]]		[[Category:Finite fields]]
	[[Category:Information theory]]		[[Category:Information theory]]


	{{telecommunications-stub}}		{{telecommunications-stub}}

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2023-11-07T22:58:21Z

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← Previous revision		Revision as of 22:58, 7 November 2023
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	The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747\|doi-access=free }}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>

	==Coding and decoding==		==Coding and decoding==

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2023-09-09T10:58:46Z

v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation)

← Previous revision		Revision as of 10:58, 9 September 2023
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	The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]]<ref name="journal-article"/>.		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]].<ref name="journal-article"/>

	==Coding and decoding==		==Coding and decoding==

2001:56A:E8A3:3A00:484F:F60B:8C2A:AC50: Clarification about the main contribution of Triangular coding.

2023-09-09T01:22:03Z

Clarification about the main contribution of Triangular coding.

← Previous revision		Revision as of 01:22, 9 September 2023
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	Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.		Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.

	~~Triangular~~ network coding ~~therefore~~ ~~essentially~~ ~~addresses~~ the ~~high encoding and~~ decoding computational complexity without degrading the throughput performance, with [[code rate]] comparable to that of ~~linear~~ ~~network~~ ~~coding~~.		The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of <math>O(n^3)</math> to <math>O(n^2)</math> (where ''n'' is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with [[code rate]] comparable to that of optimal coding schemes.

	Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]]<ref name="journal-article"/>.		Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]]<ref name="journal-article"/>.

2001:56A:E8A3:3A00:D80:2829:5D2E:C682: Added result from a recently published journal paper about Triangular codes.

2023-09-06T23:51:30Z

Added result from a recently published journal paper about Triangular codes.

← Previous revision		Revision as of 23:51, 6 September 2023
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	In [[coding theory]], '''triangular network coding''' ('''TNC''') is a [[network coding]] based packet coding scheme introduced by {{harvtxt\|Qureshi\|Foh\|Cai\|2012}}.<ref>{{Cite book\| last1 = Qureshi \| first1 = Jalaluddin		In [[coding theory]], '''triangular network coding''' ('''TNC''') is a non-linear [[network coding]] based packet coding scheme introduced by {{harvtxt\|Qureshi\|Foh\|Cai\|2012}}.<ref>{{Cite book\| last1 = Qureshi \| first1 = Jalaluddin
	\| last2 = Foh \| first2 = Chuan Heng		\| last2 = Foh \| first2 = Chuan Heng
	\| last3 = Cai \| first3 = Jianfei		\| last3 = Cai \| first3 = Jianfei
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	Triangular network coding therefore essentially addresses the high encoding and decoding computational complexity without degrading the throughput performance, with [[code rate]] comparable to that of linear network coding.		Triangular network coding therefore essentially addresses the high encoding and decoding computational complexity without degrading the throughput performance, with [[code rate]] comparable to that of linear network coding.

			Triangular code has also been proposed as [[Fountain code]]<ref name="journal-article">{{cite journal \|last1=Qureshi \|first1=Jalaluddin \|title=Triangular code: Near-optimal linear time fountain code \|journal=Digital Communications and Networks \|date=August 2023 \|volume=9 \|issue=4 \|pages=869-878 \|doi=10.1016/j.dcan.2022.12.006 \|url=https://www.sciencedirect.com/science/article/pii/S2352864822002747}}</ref> to achieve near-optimal performance with encoding and decoding computational complexity of <math>O(n\log n)</math>. It has been further shown that triangular based fountain code can even outperform optimized [[Luby transform code]]<ref name="journal-article"/>.

	==Coding and decoding==		==Coding and decoding==
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	[[File:TNC, coding 4 packets together..PNG\|frame\|An example of coding four packets using TNC. Bit ''b''<sub>''i'',''k''</sub> ∈ {0,1} is the ''i''<sup>th</sup> bit of the ''k''<sup>th</sup> packet. Each packet has original length of ''B'' bits. The resulting coded packet has length ''B'' + 3 bits. Information about the number of redundant '0' bits added at the head of each packet is included in the coded packet's header.]]		[[File:TNC, coding 4 packets together..PNG\|frame\|An example of coding four packets using TNC. Bit ''b''<sub>''i'',''k''</sub> ∈ {0,1} is the ''i''<sup>th</sup> bit of the ''k''<sup>th</sup> packet. Each packet has original length of ''B'' bits. The resulting coded packet has length ''B'' + 3 bits. Information about the number of redundant '0' bits added at the head of each packet is included in the coded packet's header.]]

	In TNC, coding is performed in two stages. First redundant "0" bits are ~~selectively~~ added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are [[Exclusive or\|XOR coded]], bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a [[Triangular matrix\|triangular pattern]].		In TNC, coding is performed in two stages. First redundant "0" bits are added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are [[Exclusive or\|XOR coded]], bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a [[Triangular matrix\|triangular pattern]].

	In essence, the TNC decoding process, like the LNC decoding process involves [[Gaussian elimination]]. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of ''triangularization,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with complexity of <math>O(n^3)</math>, where <math>n</math> is the number of packets, can be bypassed. The receiver now only needs to perform ''back-substitution,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with complexity given as <math>O(n^2)</math> for each bit location.		In essence, the TNC decoding process, like the LNC decoding process involves [[Gaussian elimination]]. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of ''triangularization,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with complexity of <math>O(n^3)</math>, where <math>n</math> is the number of packets, can be bypassed. The receiver now only needs to perform ''back-substitution,''<ref name=fraleigh95>J. B. Fraleigh, and R. A. Beauregard, Linear Algebra. Chapter 10, Addison-Wesley Publishing Company, 1995.</ref> with worst-case complexity given as <math>O(n^2)</math> for each bit location.

	==References==		==References==

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	\| doi = 10.1109/SECON.2012.6275780		\| doi = 10.1109/SECON.2012.6275780
	\| mr =		\| mr =
	\| issue =
	\| journal = IEEE Secon
	\| pages = 134–142		\| pages = 134–142
	\| volume =		\| volume =

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← Previous revision		Revision as of 02:47, 26 July 2023
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	In [[coding theory]], '''triangular network coding''' ('''TNC''') is a [[network coding]] based packet coding scheme introduced by {{harvtxt\|Qureshi\|Foh\|Cai\|2012}}.<ref>{{~~citation~~		In [[coding theory]], '''triangular network coding''' ('''TNC''') is a [[network coding]] based packet coding scheme introduced by {{harvtxt\|Qureshi\|Foh\|Cai\|2012}}.<ref>{{Cite book\| last1 = Qureshi \| first1 = Jalaluddin
	\| last1 = Qureshi \| first1 = Jalaluddin
	\| last2 = Foh \| first2 = Chuan Heng		\| last2 = Foh \| first2 = Chuan Heng
	\| last3 = Cai \| first3 = Jianfei		\| last3 = Cai \| first3 = Jianfei
			\| title = 2012 9th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON)
		⚫	\| chapter = Optimal solution for the index coding problem using network coding over GF(2)
	\| doi = 10.1109/SECON.2012.6275780		\| doi = 10.1109/SECON.2012.6275780
	\| mr =		\| mr =
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	\| journal = IEEE Secon		\| journal = IEEE Secon
	\| pages = 134–142		\| pages = 134–142
⚫	\| ~~title~~ = Optimal ~~Solution~~ for the ~~Index~~ ~~Coding~~ ~~Problem~~ ~~Using~~ ~~Network~~ ~~Coding~~ over GF(2)
	\| volume =		\| volume =
	\| year = 2012\| arxiv = 1209.6539\| bibcode = 2012arXiv1209.6539Q		\| year = 2012\| arxiv = 1209.6539\| bibcode = 2012arXiv1209.6539Q
	\| isbn = 978-1-4673-1905-8		\| isbn = 978-1-4673-1905-8
			\| s2cid = 8977891
	}}.</ref>		}}.</ref>
	Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.		Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.

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← Previous revision		Revision as of 17:55, 4 April 2020
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	\| mr =		\| mr =
	\| issue =		\| issue =
	\| journal = IEEE ~~SECON~~		\| journal = IEEE Secon
	\| pages = 134–142		\| pages = 134–142
	\| title = Optimal Solution for the Index Coding Problem Using Network Coding over GF(2)		\| title = Optimal Solution for the Index Coding Problem Using Network Coding over GF(2)
	\| volume =		\| volume =
	\| year = 2012\| arxiv = 1209.6539}}.~~</ref>~~		\| year = 2012\| arxiv = 1209.6539\| bibcode = 2012arXiv1209.6539Q
			\| isbn = 978-1-4673-1905-8
			}}.</ref>
	Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.		Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large [[finite field]] is that it resulted in high encoding and decoding [[Big O notation\|computational complexity]]. While linear encoding and decoding over [[GF(2)]] alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.