Sparse superposition codes were originally proposed as a capacity-achieving communication scheme over the gaussian channel, whose coding matrices were made of i.i.d. gaussian entries.We extend this coding scheme to more generic ensembles of rotational invariant coding matrices with arbitrary spectrum, which include the gaussian ensemble as a special case. We further introduce and analyse a decoder based on vector approximate message-passing (VAMP).Our main findings, based on both a standard replica symmetric potential theory and state evolution analysis, are the superiority of certain structured ensembles of coding matrices (such as partial row-orthogonal) when compared to i.i.d. matrices, as well as a spectrum-independent upper bound on VAMP's threshold. Most importantly, we derive a simple "spectral criterion " for the scheme to be at the same time capacity-achieving while having the best possible algorithmic threshold, in the "large section size" asymptotic limit. Our results therefore provide practical design principles for the coding matrices in this promising communication scheme.

翻译：粗略的叠加码代码最初是作为一个能力实现的通信计划而提出的,它编码矩阵是用i.i.d.gaussian条目制成的。我们把这个编码方案推广到任意频谱的旋转变量编码矩阵的更为通用的组合,其中包括千兆字节组合,作为一个特例。我们进一步引入和分析一个基于矢量近似信息传递(VAMP)的解码器。我们根据一种标准复制对称潜在理论和国家进化分析得出的主要结论,是某些结构化的编码矩阵组合(如部分行-正方形矩阵)的优越性。与i.i.d.d矩阵相比,以及一个在VAMP阈值上依赖频谱的上限。最重要的是,我们在“大段大小”的“最大节码”限制下,为这个计划同时实现能力,我们得出一个简单的“光谱标准”。因此,我们的结果为这一有希望的通信计划中的编码矩阵提供了实用的设计原则。