A note on edge fault tolerance with respect to hypercubes Original Research Article

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