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 \(2/d\). 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.