A note on edge fault tolerance with respect to hypercubes

In the previous studies on k-edge fault tolerance with respect to hypercubes Qn, matrices for generating linear k-EFT(Qn) graphs were used. Let EFTL(n,k) denote the set of matrices that generate linear k-EFT(Qn) graphs. A matrix in EFTL(n,k) with the smallest number of rows among all matrices in EFTL(n,k) is optimal. We use eftL(n,k) to denote the difference between the number of rows and the number of columns in any optimal EFTL(n,k) matrix. In terms of Hamming weight, in this work we present a necessary and sufficient condition for those matrices in EFTL(n,k) and another necessary and sufficient condition for those matrices in EFTL(n,k) of the form [InD]. We also prove that eftL(n,k+1)≥eftL(n,k)+1 and that eftL(n,k+1)=eftL(n,k)+1 if k is even.