Ring Compute-and-Forward over Block-Fading Channels

Abstract: The Compute-and-Forward protocol in quasi-static channels normally employs lattice codes based on the rational integers $\mathbb{Z}$, Gaussian integers $\mathbb{Z}\left[i\right]$ or Eisenstein integers $\mathbb{Z}\left[\omega\right]$, while its extension to more general channels often assumes channel state information at transmitters (CSIT). In this paper, we propose a novel scheme for Compute-and-Forward in block-fading channels without CSIT, which is referred to as Ring Compute-and-Forward because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers. Thanks 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 Compute-and-Forward based on $\mathbb{Z}$-lattices. By investigating the effect of Diophantine approximation by algebraic conjugates, we prove that the degrees-of-freedom (DoF) of the optimized computation rate is ${n}/{L}$, where $n$ is the number of blocks and $L$ is the number of users.