Overlapped-repetition Shor codes achieving fourfold asymptotic rate

The standard Shor code employs two repetition codes as inner and outer codes, yielding a simple structure but a relatively low code rate. By overlapping a small number of repetition codes, we enhance the asymptotic code rate fourfold. In the minimal-distance case $d = 3$, this construction reduces the overhead from $[[9,1,3]]$ to the more efficient $[[7,1,3]]$ configuration.