2. A Capacity Theorem for Lattice Codes on Gaussian Channels

A capacity theorem for lattice codes signaling is presented, which is based on an upper bound on the error probability introduced by R. de Buda.
It is shown that lattice codes can be used to achieve the channel capacity for any signal-to-noise ratio (positive statement) and the
negative statement of the capacity theorem is also proved. The sphere hardening is shown to result from the weak law of large numbers.
The proof allows a better understanding of the application of dense lattice as an efficient signaling alphabet.
An expression of the reliability function E(R,C) for lattices in AWGN channels is also presented.