We study the evolution of robustness in digital organisms adapting to a high mutation rate. As genomes adjust to the harsh mutational environment, the mean effect of single mutations decreases, up until the point where a sizable fraction (up to 30% in many cases) of the mutations are neutral. We correlate the changes in robustness along the line of descent to changes in directional epistasis, and find that increased robustness is achieved by moving from antagonistic epistasis between mutations towards codes where mutations are, on average, independent. We interpret this recoding as a breakup of linkage between vital sections of the genome, up to the point where instructions are maximally independent of each other. While such a recoding often requires sacrificing some replication speed, it is the best strategy for withstanding high rates of mutation.