Plain language

What this result means

Quantum LDPC distance estimates are often produced by heuristic decoders. Exact certification is useful because it separates a proved distance from a decoder estimate. Here the risk was overstatement, but the accessible published coprime BB/GB rows held.

  • The campaign found no overstated distances in this target set.
  • For 19 codes, CP-SAT proves the exact distance equals the published distance.
  • For nine harder codes, the report gives only the proven lower bound and the best logical operator found.

Visual notes

How to read the result

Bar chart showing nineteen confirmed qLDPC distances, nine timeout bound-only cases, and zero falsified distances.
Certificate statusThe exact solver confirmed 19 published distances, timed out on 9 larger cases, and found zero lower-distance corrections.
Scatter plot of confirmed qLDPC codes by block length and certified minimum distance.
Distance ladderEach point is a published coprime BB/GB code whose BP-OSD distance estimate was proved exact.

Result table

Nineteen published qLDPC distances certified exactly; zero overstatements found.

CellBaselineNumaroDeltaNote
[[18,4,4]]4 BP-OSD4 exactconfirmedBB family
[[48,4,8]]8 BP-OSD8 exactconfirmedGB family
[[66,4,10]]10 BP-OSD10 exactconfirmedGB family
[[196,18,8]]8 BP-OSD8 exactconfirmedBB family
9 harder codespublishedbounds onlyuncertifiedno overstatement evidence

Method

How it was found

The campaign rebuilt each CSS code from published construction polynomials and solved the minimum logical-operator problem exactly where feasible.

  • Implemented GF(2) linear algebra for CSS validity, ranks, and logical spaces.
  • Calibrated on textbook codes and known qLDPC anchors.
  • Modeled minimum-weight logical search with CP-SAT.
  • Separated exact certificates from timed-out bound-only cases.

Verification

How it was checked

verify.py rebuilds H_X and H_Z, checks CSS validity and parameters, reruns the distance proof, and re-exhibits each witnessing logical operator.

Scope

What is not being claimed

This is not a new-code result and not a record beat. It confirms 19 distances and reports bounds for unresolved larger cases.

References

Baseline sources

Citation

How to cite

Numaro Autoresearch Team. "qLDPC distances: exact certificates, zero corrections." Numaro Research Report NUMARO-2026-013, 2026.

@techreport{numaro2026QldpcDistanceCertificates,
  title = {qLDPC distances: exact certificates, zero corrections},
  author = {Numaro Autoresearch Team},
  institution = {Numaro},
  number = {NUMARO-2026-013},
  year = {2026},
  url = {https://numaro.tech/research/qldpc-distance-certificates-2026/}
}