In order to overcome the limitations imposed by DNA barcoding when multiplexing a large number of samples in the current generation of high-throughput sequencing instruments, we have recently proposed a new protocol that leverages advances in combinatorial pooling design (group testing) doi:10.1371/journal.pcbi.1003010. We have also demonstrated how this new protocol would enable de novo selective sequencing and assembly of large, highly-repetitive genomes. Here we address the problem of decoding pooled sequenced data obtained from such a protocol. Our algorithm employs a synergistic combination of ideas from compressed sensing and the decoding of error-correcting codes. Experimental results on synthetic data for the rice genome and real data for the barley genome show that our novel decoding algorithm enables significantly higher quality assemblies than the previous approach.