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

Table 3

Lower bounds for success probability of randomized coding and the required field size. Upper table shows the success probability, that is, the probability of having successful transmission with coding redundancy (𝛿_𝑡=𝑚𝑖𝑛𝑐𝑢𝑡(𝑡)−𝑑𝑖𝑚(𝒞), for all 𝑡∈𝑇) with and without degradation in the network up to 𝑑 [43]. The lower table shows the success probability of generating a minimum distance separable (MDS) code with distance 𝐷_{𝑚𝑖𝑛,𝑡}=𝑚𝑖𝑛𝑐𝑢𝑡(𝑡)−𝑑𝑖𝑚(𝒞)+1, with and without codebook redundancy [44]. 𝐽 is the set of internal nodes.

	Without degradation (𝑑=0)	With degradation (𝑑>0)
Pr(𝐷min,𝑡≤𝛿𝑡+1−𝑑,∀𝑡∈𝑇)	1−∑𝑡∈𝑇^𝛿𝑡+|𝐽|+1|𝐽|/(𝑞−1)^𝛿𝑡+1	1−∑𝑡∈𝑇𝛿^|𝐸|𝑡−𝑑^𝑑+|𝐽|+1|𝐽|/(𝑞−1)^𝑑+1	[43]
Field size 𝑞	∑𝑡∈𝑇𝛿^|𝐸|𝑡	1−(∑𝑡∈𝑇𝛿^|𝐸|𝑡−𝑑^𝑑+|𝐽|+1|𝐽|)^1/𝑑+1
	Without redundancy (𝛿𝑡=0)	With redundancy (𝛿𝑡>0)
MDS, Pr(𝐷min,𝑡≤𝛿𝑡+1,∀𝑡∈𝑇)	(1−|𝑇|/(𝑞−1))^|𝐽|+1	(1−∑𝑡∈𝑇𝛿^|𝐸|𝑡/(𝑞−1))^|𝐽|+1	[44]
Field size 𝑞	|𝑇||𝐸|	∑𝑡∈𝑇𝛿^|𝐸|𝑡