which transmits $X_j^n = \sum_{i=1}^N X_{ji}^n$. Also, each transmitter assigns a power $P_i$ for encoding its $i^{th}$ sub-message. Note that the subscript in rate and power does not indicate user, but the sub-messages. The power $P_i$ is chosen as $$\label{eqn : power assignment 1}
P_i = (a^2 - 1)(2a^4-a^2)^{N-i}, \ \ i\in\{1,2,\ldots,N\}.$$ We explain the encoding and decoding strategy below in detail.

Encoding Strategy: Each transmitter encodes all its sub-messages using lattice coding, and chooses lattices $\Lambda_1, \ldots, \Lambda_N$, shifts $s_1, \ldots, s_N$ and spherical shaping regions $S_1, \ldots, S_N$. The codebook for $i^{th}$ sub-message at each transmitters is denoted by $\mathcal{C}_i = (\Lambda_i + s_1) \cap S_i$. Decoding Strategy: The received signal at receiver $j$ is $$Y_j^n = \sum_{i=1}^N X_{ji}^n + \sum_{k=1, k \neq j}^3\sum_{i=1}^N a X_{ki}^n + Z_j^n.$$ We denote the interference at receiver $j$ due to the $i^{th}$ sub-message from the other transmitters by $I_{ji}^n$ given by $$I_{ji}^n = \sum_{k=1, k \neq j}^3 a X_{ki}^n.$$ The decoding process at receiver $j$ proceeds through $N$ stages. At stage $i$, receiver $j$ first decodes interference $I_{ji}^n$ and then decodes message $m_{ji}$. In decoding the interference $I_{ji}^n$, receiver $j$ sees interference plus noise of $$\sum_{k=i}^{N} X_{jk}^n + \sum_{l=1,l\neq j}^{3}\sum_{k=i+1}^{N} a 