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In most of the network coding problems with $k$ messages, the existence ofbinary network coding solution over $\mathbb{F}_2$ depends on the existence ofadequate sets of $k$-dimensional binary vectors such that each set comprises oflinearly independent vectors. In a given $k \times n$ ($n \geq k$) binarymatrix, there exist ${n}\choose{k}$ binary sub-matrices of size $k \times k$.Every possible $k \times k$ sub-matrix may be of full rank or singulardepending on the columns present in the matrix. In this work, for full rankbinary matrix $\mathbf{G}$ of size $k \times n$ satisfying certain condition onminimum Hamming weight, we establish a relation between the number of full ranksub-matrices of size $k \times k$ and the weight enumerating function of theerror correcting code with $\mathbf{G}$ as the generator matrix. We give analgorithm to compute the number of full rank $k \times k$ submatrices.

A Relation Between Weight Enumerating Function and Number of Full Rank Sub-matrices