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 ℤ, the Gaussian integers ℤ[i], or the Eisenstein integers ℤ[ω], 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 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.