Motivated by hyper-reliable low-latency communication in 6G, we consider error control coding for short block lengths in multi-antenna SIMO flat-fading channels. In general, the channel fading coefficients are unknown at both the transmitter and receiver, which is referred to as a non-coherent channel. Conventionally, pilot symbols are transmitted to facilitate channel estimation, causing power and bandwidth overhead. Our paper considers sparse regression codes (SPARCs) for non-coherent flat-fading channels without using pilots. We develop a novel greedy decoder for SPARC using maximum likelihood (ML) principles, referred to as maximum likelihood matching pursuit (MLMP). MLMP works based on successive combining principles as opposed to conventional greedy algorithms, which are based on successive cancellation. We also obtain the noiseless perfect recovery condition for our successive combining algorithm. We further introduce an enhanced version named parallel-MLMP (P-MLMP) to improve performance by mitigating error propagation in greedy methods. In addition, we develop an approximate message passing (AMP) SPARC decoder for the non-coherent SIMO flat-fading model. Using simulation studies, we show that the MLMP decoder for SPARC outperforms AMP and other greedy decoders. Also, SPARC with P-MLMP decoder outperforms polar codes employing pilot-based channel estimation and polar codes with non-coherent decoders.

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.