Singularity Probability Analysis for Sparse Random Linear Network Coding

Motivated by the noncoherent subspace coding approach and the low-complexity sparse coding approach to realize random linear network coding, we 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. We 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. This result provides insight for optimizing sparse random linear network coding schemes and allows the problem of determining the probability of having a nonsingular transfer matrix over a large field size to be transformed into a combinatorial problem. By using some combinatorial arguments, useful upper and lower bounds on the singularity probability of the random transfer matrix are derived.