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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
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| | last1 = Qureshi | first1 = Jalaluddin
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| | last2 = Foh | first2 = Chuan Heng
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| | last3 = Cai | first3 = Jianfei
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| | doi = 10.1109/SECON.2012.6275780
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| | issue =
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| | journal = IEEE SECON
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| | pages = 134–142
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| | title = Optimal Solution for the Index Coding Problem Using Network Coding over GF(2)
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| | volume =
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| | year = 2012}}.</ref>
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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.
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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.
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| ==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.]]
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| 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]].
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| 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.
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| ==References==
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| {{reflist}}
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| [[Category:Coding theory]]
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| [[Category:Finite fields]]
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| [[Category:Information theory]]
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| {{telecommunications-stub}}
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