We propose a new compression scheme for genomic data given as sequence fragments called reads. The scheme uses a reference genome at the decoder side only, freeing the encoder from the burdens of storing references and performing computationally costly alignment operations. The main ingredient of the scheme is a multi-layer code construction, delivering to the decoder sufficient information to align the reads, correct their differences from the reference, validate their reconstruction, and correct reconstruction errors. The core of the method is the well-known concept of distributed source coding with decoder side information, fortified by a generalized-concatenation code construction enabling efficient embedding of all the information needed for reliable reconstruction. We first present the scheme for the case of substitution errors only between the reads and the reference, and then extend it to support reads with a single deletion and multiple substitutions. A central tool in this extension is a new distance metric that is shown analytically to improve alignment performance over existing distance metrics.