基于乘积构造的多级格码的多级计算与转发
Multistage Compute-and-Forward with Multilevel Lattice Codes Based on Product Constructions
摘要：
提出了一种新的格结构。这种构造可以被认为是构造A，其代码可以分别表示为$L$线性码在$\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$上的笛卡尔乘积;因此，被称为乘积构造。示出了这样的格，是好的量化和Poltyev良好的多级解码下的序列的存在。然后使用这个格族来生成嵌套格码序列，其允许在多级解码下实现与Nazer和Gastpar相同的计算速率用于计算和转发，这被称为基于格的多级计算和转发。受所提出的格码的启发，两个家庭的信号星座，然后提出了分离为基础的计算和转发的框架提出的Tunali \textit{et al.}连同一个多级编码/多级解码方案，专门为这些星座。该方案被称为基于分离的多级计算和转发，并且被示出具有由星座大小的最大公约数（可以不是素数）而不是星座大小本身支配的信道编码的复杂度。
英文摘要：
A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of $L$ linear codes over $\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward. Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by Tunali \textit{et al.} together with a multilevel coding/multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself.
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