Sparse Regression Code (SPARC) connects the sparse signal recovery framework with the error control coding techniques. In this paper, we focus on improving the block error performance of SPARC in short block length regime over the AWGN channel. Towards that, we introduce suitable candidates for dictionary matrices in real and complex fields using Gold sequences and mutually unbiased bases (MUB). We propose two generalizations of SPARC (GSPARC), develop a greedy decoder called Match and Decode (MAD) algorithm, and provide its analytical noiseless recovery guarantees. We propose a parallel greedy search technique called parallel MAD (PMAD) to improve 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 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 with the same spectral efficiency.

Inspired by compressive sensing principles, we propose novel error control coding techniques for communication systems. The information bits are encoded in the support and the non-zero entries of a sparse signal. By selecting a dictionary matrix with suitable dimensions, the codeword for transmission is obtained by multiplying the dictionary matrix with the sparse signal. Specifically, the codewords are obtained from the sparse linear combinations of the columns of the dictionary matrix. At the decoder, we employ variations of greedy sparse signal recovery algorithms. Using Gold code sequences and mutually unbiased bases from quantum information theory as dictionary matrices, we study the block error rate (BLER) performance of the proposed scheme in the AWGN channel. Our results show that the proposed scheme has a comparable and competitive performance with respect to the several widely used linear codes, for very small to moderate block lengths. In addition, our coding scheme extends straightforwardly to multi-user scenarios such as multiple access channel, broadcast channel, and interference channel. In these multi-user channels, if the users are grouped such that they have similar channel gains and noise levels, the overall BLER performance of our proposed scheme will coincide with an equivalent single-user scenario.