On Scaling Rules for Energy of VLSI Polar Encoders and Decoders

It is shown that all polar encoding schemes of rate R > 1 2 of block length N implemented according to the Thompson VLSI model must take energy E ≥ Ω ( N ) . This lower bound is achievable up to polylogarithmic factors using a mesh network topology defined by Thompson and the encoding algorithm defined by Arikan. A general class of circuits that compute successive cancellation decoding adapted from Arikan’s butterfly network algorithm is defined. It is shown that such decoders implemented on a rectangle grid for codes of rate R > 2/3 must take energy E ≥ Ω(N), and this can also be reached up to polylogarithmic factors using a mesh network. Capacity approaching sequences of energy optimal polar encoders and decoders, as a function of reciprocal gap to capacity χ = (1 −R/C)−1, have energy that scales as Ω ( χ ) ≤ E ≤ O ( χ log (χ) ) .