for sufficiently large $p$ and small $\gamma$ such that $\gamma^n p^{n-k} = V$. Hence, the probability of decoding error for at least three fourths of the lattices in $\mathcal{L}_{B}$ satisfies $$\label{eqn : Loeliger condition}
Pe < 4(1 + \delta) \frac{2^{n \frac{1}{2}\log(2\pi e \sigma^2)}}{V}.$$

The proof of the lemma is described in [@Loeliger] and is omitted here. The next lemma describes the existence of a good lattice code for a single user AWGN channel.

Consider a single user point to point additive noise channel described in (\[eqn : Gaussian channel\]) such that the transmitter satisfies a power constraint of $P$. Then, we can choose a lattice $\Lambda$ generated using construction A, a shift $s$ and a shaping region $S$ such that the codebook $(\Lambda + s) \cap S$ achieves a rate $R$ with arbitrarily small probability of error if $$R \leq \frac{1}{2}\log\left(\frac{P}{\sigma^2}\right).$$

The proof of the lemma is described in [@Loeliger]. In essence, the above lemma describes the existence of a lattice code with sufficient codewords. It should be noted that lattice coding with a random dither has been used in [@ErezZamir2004] to achieve the full capacity of the AWGN channel. In the next section, we describe a layered lattice coding 