the cardinality of the codebook $\mathcal{C}_i$ satisfies $|\mathcal{C}_i| = |(\Lambda_i + s_i) \cap S_i| \geq 2^{nR_i}$. Next, we describe the probability of error analysis and rate constraints at receiver $1$. The analysis and the rate constraints at other receivers are the same.

Receiver $1$ first decodes interference $I_{11}^n$ and message $m_{11}$. The interference plus noise power when decoding $I_{11}^n$ is given by $P_1 + \sum_{k=2}^N (2a^2 + 1)P_k + 1$. With the choice of lattice $\Lambda_1$, the probability of decoding error is upper bounded by $$P_{e1}^{int} \leq 4(1 + \delta) \frac{2^{n \frac{1}{2}\log(2\pi e (P_1 + \sum_{k=2}^N (2a^2 + 1)P_k + 1))}}{a^nV_1},$$ where $a^n V_1$ is the volume of the Voronoi region of the lattice $a \Lambda_1$ (the interference lattice of message $m_{21}$ and $m_{31}$). Hence, the probability of error decays with $n$ if $$R_1 \leq \frac{1}{2}\log\left(\frac{a^2 P_1}{P_1 + \sum_{k=2}^N (2a^2 + 1)P_k + 1}\right).$$ Similarly, in decoding the message $m_{11}$, the interference plus noise power seen by receiver $1$ is equal to $\sum_{k=2}^N (2a^2 + 1)P_k + 1$. The probability of decoding error is upper bounded by $$P_{e1}^{message} \leq 4(1 + \delta) \frac{2^{n \frac{1}{2}\log(2\pi e (\sum_{k=2}^N (2a^2 + 1)P_k + 1))}}{V_1}.$$ Hence, the probability of error decays with $n$ if 