Energy of decoding algorithms

A standard model for the energy of a computation was developed by Thompson, which relates the energy of a computation to the product of the area a circuit and the number of clock cycles needed to carry out the computation. We show that for any circuit implemented this way, using any algorithm that performs decoding of a codeword passed through a binary symmetric channel, as the block length approaches infinity either (a) the probability of block error approaches 1 or (b) the energy of the computation scales at least as equation. This implies that the energy of successful decoding, per decoded bit, must scale at least as equation.