A Relation Between Weight Enumerating Function and Number of Full Rank Sub-matrices
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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 nk binary sub-matrices of size k × k. Every possible k × k sub-matrix 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.
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