Improved Expression for Rank Distribution of Sparse Random Linear Network Coding

Characterization of the rank distribution of a sparse random matrix over a finite field can be decomposed into two subproblems. The first subproblem is the characterization of the probability p(i,n) that an n-dimensional vector is linearly dependent of other i linearly independent n-dimensional vectors. The second subproblem is the characterization of the rank distribution of a sparse random matrix as a function of p(i,n). In this letter, we focus on the second subproblem and present an exact solution to it. The derivation is based on an absorbing Markov chain and the eigen decomposition of the transition matrix. Compared with the state-of-the-art expressions, the derived expression is closed-form and of lower complexity. As a necessity for the exactness of the derived expression, we prove that the derived expression is equivalent to the state-of-the-art expressions.