Current algorithms for compressing genomic data mostly focus on achieving high levels of effectiveness and reasonable levels of efficiency, ignoring the need for features such as random access and stream processing. Therefore, in this paper, we introduce a novel framework for compressing genomic data, with the aim of allowing for a better trade-off between effectiveness, efficiency and functionality. To that end, we draw upon concepts taken from the area of media data processing. In particular, we propose to compress genomic data as small blocks of data, using encoding tools that predict the nucleotides and that correct the prediction made by storing a residue. We also propose two techniques that facilitate random access. Our experimental results demonstrate that the compression effectiveness of the proposed approach is up to 1.91 bits per nucleotide, which is significantly better than binary encoding (3 bits per nucleotide) and Huffman coding (2.21 bits per nucleotide).