A Survey of Linear Network Coding and Network Error Correction Code Constructions and Algorithms

Table 4

Comparison of centralized algorithms for single-source multicast network error correction code construction. |𝐸| is the total number of edges in the network, |𝑇| the number of sinks, 𝑑 is the minimum distance at all sinks, ℎ is the multicast rate, 𝑛_𝑠 the number of outgoing channels from the source. In Bahramgiri and Lahouti's algorithms: 𝑅^(𝑑) is the set of failure patterns with at most 𝑑−1 failures of edges in common with other paths, Ω includes all possible sets of paths [55].

	Field size (q)	Time complexity
Yang et al. [50], Algorithm 1	|𝑇|^|𝐸|𝑑−1	𝒪(𝑛𝑠|𝑇|𝜔^|𝐸|𝑑−1(𝑛^2𝑠+|𝑇|^|𝐸|𝑑−1))
Yang et al. [50], Algorithm 2	|𝑇|^ℎ+|𝐸|−2𝑑−1	𝒪(ℎ|𝑇|^2|𝐸|^ℎ+|𝐸|−2𝑑−1(ℎ^2|𝑇|^2+|𝑇|^ℎ+|𝐸|−2𝑑−1)+ℎ^3𝑑|𝐸||𝑇|^|𝐸|𝑑−1)
Matsumoto [56]	|𝑇|^|𝐸|𝑑−1	𝒪(ℎ|𝐸||𝑇|^|𝐸|𝑑−1(|𝑇|^|𝐸|𝑑−1+ℎ+𝑑))
Bahramgiri and Lahouti [55], RNC1	|𝑇|^ℎ−1ℎ−𝑑	𝒪(|𝐸||𝑇|(𝑛−𝑑+1)^3ℎ−1ℎ−𝑑(1−^ℎ−1ℎ−𝑑|𝑇|/𝑞)^−1)
Bahramgiri and Lahouti [55], RNC2	|𝑇||𝑅^(𝑑)|^ℎ−1ℎ−𝑑	𝒪(|𝐸||𝑇||𝑅^(𝑑)|(𝑛−𝑑+1)^3ℎ−1ℎ−𝑑(1−^ℎ−1ℎ−𝑑|𝑇||𝑅^(𝑑)|/𝑞)^−1)
Bahramgiri and Lahouti [55], RNC3	|𝑇|∏𝑇|Ω|	𝒪(|𝐸||𝑇|(𝑛−𝑑+1)^3∏𝑇|Ω|(1−|𝑇|∏𝑇|Ω|/𝑞)^−1)
Guang et al. [57], lin. ind. test	|𝑇|^|𝐸|𝑑−1	𝒪(|𝐸|(𝜔+𝑑−1)|𝑇|^|𝐸|𝑑−1(1+𝜔+(|𝐸|+1)/2))
Guang et al. [57], deterministic	|𝑇|^|𝐸|𝑑−1	𝒪(|𝐸||𝑇|^|𝐸|𝑑−1(|𝑇|^|𝐸|𝑑−1(𝜔+(|𝐸|+1)/2)+𝜔+𝑑−1))