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New results on low-density integer lattices

机译：低密度整数晶格上的新结果

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A new family of integer lattices built from Construction A and non-binary low-density parity-check (LDPC) codes has been proposed by the authors in 2012. Lattices in this family are referred to as LDA lattices. Previous experimental results revealed excellent performance which clearly single out LDA lattices among the strongest candidates for potential applications in digital communications and networks, such as network coding and information theoretic security at the physical layer level. In this paper, we show that replacing random codes by LDPC codes in Construction A does not induce any structural loss. More precisely, our main theorem states that LDA lattices can achieve Poltyrev capacity limit on an additive white Gaussian noise channel. We present here the detailed proof and its consequences on the lattice dimension, the finite field size, and the parameters of the LDPC ensemble. The latter has a row weight that increases logarithmically in the code length. In a more recent work, it is proved that the Poltyrev limit is attained by a different LDA ensemble having a small constant row weight.
机译：作者在2012年提出了一种新的由构造A和非二进制低密度奇偶校验（LDPC）码构建的整数晶格族。该族中的晶格称为LDA晶格。先前的实验结果表明，出色的性能可以清楚地将LDA晶格选出为数字通信和网络中潜在应用的最强候选者，例如物理层级别的网络编码和信息理论安全性。在本文中，我们表明在结构A中用LDPC码替换随机码不会引起任何结构损失。更准确地说，我们的主要定理指出，LDA晶格可以在加性高斯白噪声通道上达到Poltyrev容量极限。我们在这里提供详细的证明及其对晶格尺寸，有限域大小和LDPC集成参数的影响。后者的行权重在代码长度上呈对数增加。在最近的工作中，证明了通过具有小的恒定行权重的不同的LDA合奏来达到Poltyrev极限。

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