$$\label{eqn : lattice set}
\mathcal{L}_{B} = \{\Lambda_C : C \in \mathcal{B}\}.$$ We consider a single user point to point additive noise channel $$\label{eqn : Gaussian channel}
Y = X + Z$$ where $X$ is the transmitted signal, $Y$ the received signal and $Z$ is the additive noise of zero mean and variance equal to $\sigma^2$ that corrupts the transmitted signal at the receiver. If the transmitted word over time is a lattice point, then it can be shown that a suitable lattice and a decoding strategy exists such that the probability of decoding error can be made arbitrarily small as the number of dimensions of the lattice increases. This result is stated formally in the following lemma.

\[lem : Loeliger\] Consider a single user point to point additive noise channel described in (\[eqn : Gaussian channel\]). Let $\mathcal{B}$ be a balanced set of linear $(n,k)$ codes over $\mathbb{Z}_p$. Averaged over all lattices from the set $\mathcal{L}_{B}$ given in (\[eqn : lattice set\]), each scaled by $\gamma > 0$ and with a fundamental volume $V$, we have that for any $\delta > 0$, the average probability of decoding error is bounded by $$\overline{P_e} < (1 + \delta) \frac{2^{n \frac{1}{2}\log(2\pi e \sigma^2)}}{V}.$$ 