Motivated by hyper-reliable low-latency communication in 6G, we consider error control coding for short block lengths in multi-antenna fading channels. In general, the channel fading coefficients are unknown at both the transmitter and receiver, which is referred to as non-coherent channels. 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 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. In addition, we develop an approximate message passing (AMP) SPARC decoder for the non-coherent flat fading model. Using simulation studies, we show that the MLMP decoder for SPARC outperforms AMP and other greedy decoders. Also, SPARC with MLMP decoder outperforms polar codes employing pilot-based channel estimation and polar codes with non-coherent decoders.