A soliton distribution is a type of discrete probability distribution that arises in the theory of erasure correcting codes. A paper by Luby 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

Probability mass function

In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

 is given by

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-values 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
While the ideal soliton distribution has a mode

Mode (statistics)

In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....

(or spike) at 1, the effect of the extra component in the robust distribution is to add an additional spike at the value M.