Weight Enumerating Function, Number of Full Rank Sub-matrices and Network Coding

In most of the network coding problems with k messages, the existence of binary network coding solution over ${\mathbb{F}_2}$ depends on the existence of adequate sets of k-dimensional binary vectors such that each set comprises of linearly independent vectors. In a given k×n (n ≥ k) binary matrix, there exist $ \binom{n}{k}$ binary sub-matrices of size k×k. Every possible k×k submatrix may be of full rank or singular depending on the columns present in the matrix. In this work, for full rank binary matrix G of size k×n satisfying certain condition on minimum Hamming weight, we establish a relation between the number of full rank sub-matrices of size k×k and the weight enumerating function of the error correcting code with G as the generator matrix. We give an algorithm to compute the number of full rank k×k submatrices.