Ring Compute-and-Forward over Block-Fading Channels
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The Compute-and-Forward protocol in quasi-static channels normally employs lattice codes based on the rational integers Z, Gaussian integers Z[i] or Eisenstein integers Z[ω]. In this paper, we propose a novel scheme for Compute-and-Forward in block-fading channels, 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 Z-lattices. By investigating the effect of Diophantine approximation by algebraic conjugates, we prove that the degrees-of-freedom (DoF) of the computation rate is n/L, while the DoF of the sum-rate is n, where n is the number of blocks and L is the number of users.
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