In the previous studies on -edge fault tolerance with respect to hypercubes , matrices for generating linear graphs were used. Let denote the set of matrices that generate linear graphs. A matrix in with the smallest number of rows among all matrices in is optimal. We use to denote the difference between the number of rows and the number of columns in any optimal matrix. In terms of Hamming weight, in this work we present a necessary and sufficient condition for those matrices in and another necessary and sufficient condition for those matrices in of the form . We also prove that and that if is even.