Ring Compute-and-Forward Over Block-Fading Channels.

The compute-and-forward (C&F) protocol in quasi-static channels normally employs lattice codes based on the rational integers $\mathbb {Z}$ , the Gaussian integers $\mathbb {Z} [i]$ , or the Eisenstein integers $\mathbb {Z} [\omega ]$ , while its extension to more general channels often assumes channel state information at transmitters (CSIT). In this paper, we propose a novel scheme for C&F in block-fading channels without CSIT, which is referred to as ring C&F because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Owing to the multiplicative closure of the algebraic lattices employed, a relay is able to decode an algebraic–integer linear combination of lattice codewords. We analyze its achievable computation rates and show it outperforms conventional C&F based on the $\mathbb {Z}$ -lattices. By investigating the effect of the Diophantine approximation by algebraic conjugates, we prove that the degrees of freedom (DoFs) of the optimized computation rate are ${n}/{L}$ , where $n$ is the number of blocks and $L$ is the number of users. [ABSTRACT FROM AUTHOR]