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A '''soliton distribution''' is a type of [[discrete probability distribution]] that arises in the theory of [[erasure correcting code]]s. A paper by Luby<ref name="Luby">{{cite conference | last = Luby | first = M. | year = 2002 | url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1181950 | title = LT Codes | conference = The 43rd Annual IEEE Symposium on Foundations of Computer Science }}</ref> introduced two forms of such distributions, the '''ideal soliton distribution''' and the '''robust soliton distribution'''. | |||
==Ideal distribution== | |||
The '''ideal soliton distribution''' is a probability distribution on the integers from 1 to ''N'', where ''N'' is the single parameter of the distribution. The [[probability mass function]] is given by<ref name=T>{{cite paper | first = Tuomas | last = Tirronen | year = 2005 | id = {{citeseerx|10.1.1.140.8104}} | title = Optimal Degree Distributions for LT Codes in Small Cases | publisher = Helsinki University of Technology }}</ref> | |||
:<math> | |||
p(1)= \frac{1}{N}, </math> | |||
:<math> | |||
p(k)= \frac{1}{k(k-1)} \qquad (k=2,3,\dots,N). \, | |||
</math> | |||
==Robust distribution== | |||
The '''robust''' form of distribution is defined by adding an extra set of values to the elements of mass function of the ideal soliton distribution and then standardising so that the values add up to 1. The extra set of values, ''t'', are defined in terms of an additional real-valued parameter ''δ'' (which is interpreted as a failure probability) and an integer parameter ''M'' (''M'' < ''N'') . Define ''R'' as ''R''=''N''/''M''. Then the values added to ''p''(''i''), before the final standardisation, are<ref name=T/> | |||
:<math> | |||
t(i)= \frac{1}{iM}, \qquad \qquad (i=1,2,\dots,M-1), \, | |||
</math> | |||
:<math> | |||
t(i)= \frac{\ln(R/\delta)}{M}, \qquad (i=M), \, | |||
</math> | |||
:<math> | |||
t(i)= 0, \qquad \qquad (i=M+1,\dots,N). \, | |||
</math> | |||
While the ideal soliton distribution has a [[mode (statistics)|mode]] (or spike) at 1, the effect of the extra component in the robust distribution is to add an additional spike at the value ''M''. | |||
==See also== | |||
*[[Luby transform code]] | |||
==References== | |||
{{reflist}} | |||
{{ProbDistributions|discrete-finite}} | |||
[[Category:Discrete distributions]] | |||
[[Category:Coding theory]] | |||
[[Category:Probability distributions]] | |||
Revision as of 21:34, 10 January 2014
A soliton distribution is a type of discrete probability distribution that arises in the theory of erasure correcting codes. A paper by Luby[1] introduced two forms of such distributions, the ideal soliton distribution and the robust soliton distribution.
Ideal distribution
The ideal soliton distribution is a probability distribution on the integers from 1 to N, where N is the single parameter of the distribution. The probability mass function is given by[2]
Robust distribution
The robust form of distribution is defined by adding an extra set of values to the elements of mass function of the ideal soliton distribution and then standardising so that the values add up to 1. The extra set of values, t, are defined in terms of an additional real-valued parameter δ (which is interpreted as a failure probability) and an integer parameter M (M < N) . Define R as R=N/M. Then the values added to p(i), before the final standardisation, are[2]
While the ideal soliton distribution has a mode (or spike) at 1, the effect of the extra component in the robust distribution is to add an additional spike at the value M.
See also
References
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