Determine the local-dimension dependence of the HD-decoder threshold

Determine how the threshold of the hard-decision decoder for qudit many-hypercube codes depends on the local dimension q.

Background

The paper compares hard-decision (HD) and level-by-level minimum-distance (LLMD) decoders for qudit many-hypercube codes under an independent qudit X-error model. The LLMD threshold increases monotonically with q, whereas the corresponding numerical results for the HD decoder do not display a clear dependence on q. The unresolved issue is to characterize whether, and in what manner, the HD-decoder threshold varies with the qudit dimension.

References

In contrast, the $q$-dependence is not clear for the HD decoder.

— High-Rate Concatenated Quantum Error-Correcting Codes for Qudits  (2610.11225 - Nishi et al., 8 Oct 2026) in Section 3.1, “Code-capacity simulations”