Single-shot and two-shot decoding with generalized bicycle codes

Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance $d_{\rm S}=3$. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering $T\ge 1$ measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding ($T=1$) suffers from a significant loss of accuracy, already two-shot ($T=2$) decoding gives nearly the same logical error rates as multi-shot with much larger $T$. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all $T\ge 1$.