**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.

** **