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.

翻译：通过在少数重复块中引入受控重叠，可在保持平均稳定子权重为 \(4\) 的同时实现四倍渐近码率提升。将重叠外层替换为LDPC码可进一步构造出渐近码率为 \(2/d\) 的码族。我们还描述了一种恒定激发变体，可在不增加额外开销的情况下抑制集体相干误差，以及一种玻色子推广形式，可将该框架扩展至振荡器编码。所得码族的码率介于旋转表面码（平均稳定子权重 \(4\)）与BB码（平均稳定子权重 \(6\)）之间，同时避免了迭代解码器典型的性能退化问题。