Bounds on the expected rank of sparse linear operator channel matrices

The random linear network coding system can be modelled as a linear operator channel (LOC). Among all the LOC models, one of the most interesting models is the sparse LOC model, where the entries of the channel matrix take zero with high probability and any of the nonzero elements in the finite field with equal probabilities. It is well-known that the normalized capacity of LOC is characterized by the rank distribution of the channel matrix. Furthermore, Yang et al. derived upper and lower capacity bounds for LOC in terms of the expected rank of the channel matrix, and the two bounds coincide with the expected rank of the channel matrix as the packet size increases. Therefore, in this paper, we investigate the expected rank of the sparse LOC matrix. By calculating the number of vectors in the null space of the sparse LOC matrix, a lower bound on the expected rank of the sparse matrix is obtained. The linear rank distribution of the sparse LOC matrix is also calculated, which is then used to derive an upper bound on the expected rank of such matrices. Besides, the upper bound can be shown to be exact as the matrix size tends to infinity. Numerical results show that the bounds are rather tight for various matrix sizes and sparsity.