Singularity Probability Analysis for Sparse Random Linear Network Coding

Motivated by the sparse random linear network coding approach, the authors consider the problem of characterizing the probability of having a full rank (or nonsingular) square transfer matrix over a finite field, for which the probability of choosing the zero element is different from that of choosing a nonzero element. They found that for a sufficiently large field size, whether the transfer matrix is singular or not is determined with probability one by the zero pattern of the matrix, i.e., where the zeroes are located in the matrix.