Perfect Fault Tolerance of the n-k-n Network

It was shown in Phatak, Choi, and Koren (1993) that the neural network implementation of the n-k-n encoder-decoder has a minimal size of k = 2, and it was shown how to construct the network. A proof was given in Phatak and Koren (1995) that in order to achieve perfect fault tolerance by replicating the hidden layer, the required number of replications is at least three. In this note, exact lower bounds are derived for the number of replications as a function of n, for the n-2-n network and for the n-log₂n-n network.