Overlapped-repetition Shor codes achieving fourfold asymptotic rate

Introducing controlled overlap among a few repetition blocks yields a fourfold asymptotic rate improvement while preserving an average stabilizer weight of $4$. Substituting the overlapped outer layer with an LDPC code further produces a family of constructions with asymptotic rate $2\mathrm{/}d$. We also describe a constant-excitation variant that suppresses collective coherent errors without additional overhead, as well as a bosonic generalization that extends the framework to oscillator encodings. The resulting family achieves a code rate intermediate between that of the rotated surface code (average stabilizer weight $4$) and the BB code (average stabilizer weight $6$), while remaining free of the performance degradation typical of iterative decoders.