X_{lk}^n + Z_j^n$$ with an interference plus noise power $\leq P_i + \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$. In decoding message $m_{ji}$, receiver $j$ sees an interference plus noise $$\sum_{k=i+1}^{N} X_{jk}^n + \sum_{l=1,l\neq j}^{3}\sum_{k=i+1}^{N} a X_{lk}^n + Z_j^n$$ with an interference plus noise power $\leq \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$. Next, we describe the choice of lattices, shifts and spherical regions, before proceeding to probability of error analysis and rate constraints at the receivers.

Choice of Lattices, Shifts and Shaping Regions: For $i \in \{1,\ldots, N\}$, each transmitter chooses shaping region $S_i$ to be an $n$ dimensional sphere of radius $\sqrt{nP_i}$. The volume of the shaping region $S_i$ is denoted by $V_{S_i}$. Lattice $\Lambda_i$ is generated using construction A such that

-   the volume of the Voronoi region $V_i = 2^{-nR_i}V_{S_i}$,

-   in decoding interference $I_{ji}^n$ at receiver $j$, the probability of error is upper bounded by (\[eqn : Loeliger condition\]) with $\sigma^2 = P_i + \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$, and

-   in decoding message $m_{ji}$ at receiver $j$, the probability of error is upper bounded by (\[eqn : Loeliger condition\]) with $\sigma^2 = \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$.

Finally, shift $s_i$ is chosen such that 