growth of sum capacity in terms of $\log(\textrm{SNR})$.

Lattice Coding Preliminaries
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In this section, we review some preliminaries on lattice coding and summarize some results on lattice coding that we use in this paper. A lattice $\Lambda$ of dimension $n$ is described by $$\Lambda = \{\lambda = Gx : x \in \mathbb{Z}^n\},$$ where $G$ is the generator matrix that describes the lattice. Let $\Omega_{\Lambda}$ denote the fundamental Voronoi region of the lattice $\Lambda$ and $V_{\Lambda}$ denote the volume of $\Omega_{\Lambda}$. In this paper, we use lattices generated using construction A described below (used in [@Loeliger]).

For any positive integer $p$, let $\mathbb{Z}_p$ denote the set of integers modulo $p$. Let $g : \mathbb{Z}^n \rightarrow \mathbb{Z}_p^n$ denote the componentwise modulo $p$ operation over integer vectors. Let $C$ denote a linear $(n,k)$ code over $\mathbb{Z}_p$. Then the lattice $\Lambda_C$ given by $$\label{eqn : construction A}
\Lambda_C = \left\{v \in \mathbb{Z}^n : g(v) \in C\right\}$$ is generated using construction A with respect to the linear code $C$. A set $\mathcal{B}$ of linear codes over $\mathbb{Z}_p$ is “balanced" if every nonzero element of $\mathbb{Z}_p^n$ is contained in the same number of codes in $\mathcal{B}$. Let $\mathcal{L}_B$ be the set of lattices denoted by 