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 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 (n) 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.