Sparse regression codes (SPARC) connect the sparse signal recovery framework of compressive sensing with error control coding techniques. SPARC encoding produces codewords which are \emph{sparse} linear combinations of columns of a dictionary matrix. SPARC decoding is accomplished using sparse signal recovery algorithms. We construct dictionary matrices using Gold codes and mutually unbiased bases and develop suitable generalizations of SPARC (GSPARC). We develop a greedy decoder, referred as match and decode (MAD) algorithm and provide its analytical noiseless recovery guarantees. We propose a parallel greedy search technique, referred as parallel MAD (PMAD), to improve the performance. We describe the applicability of GSPARC with PMAD decoder for multi-user channels, providing a non-orthogonal multiple access scheme. We present numerical results comparing the block error rate (BLER) performance of the proposed algorithms for GSPARC in AWGN channels, in the short block length regime. The PMAD decoder gives better BLER than the approximate message passing decoder for SPARC. GSPARC with PMAD gives comparable and competitive BLER performance, when compared to other existing codes. In multi-user channels, GSPARC with PMAD decoder outperforms the sphere packing lower bounds of an orthogonal multiple access scheme, which has the same spectral efficiency.

翻译：稀疏回归码（SPARC）将压缩感知中的稀疏信号恢复框架与纠错编码技术相结合。SPARC编码生成的码字是字典矩阵列的稀疏线性组合，而SPARC解码则通过稀疏信号恢复算法实现。我们利用Gold码和互无偏基构造字典矩阵，并提出了SPARC的合适推广形式（GSPARC）。我们开发了一种名为匹配与解码（MAD）的贪婪解码算法，并给出了其在无噪声条件下的恢复保证。为提升性能，我们提出了并行贪婪搜索技术——并行MAD（PMAD）。我们描述了GSPARC结合PMAD解码器在多用户信道中的适用性，提供了一种非正交多址接入方案。我们通过数值结果，比较了所提算法在短码长体制下、AWGN信道中GSPARC的块错误率（BLER）性能。PMAD解码器在SPARC上取得了优于近似消息传递解码器的BLER性能。与其他现有码相比，GSPARC结合PMAD展现了可比较且具有竞争力的BLER性能。在多用户信道中，GSPARC结合PMAD解码器优于具有相同频谱效率的正交多址接入方案的球堆积下界。