Which open entries in the tables of best known linear codes over GF(3) to GF(9) can be improved, with a minimum distance anyone can recompute?
A linear code adds redundancy to data so that errors can be corrected. Its symbols come from a finite field GF(q), a number system with q elements. Here q is 3, 4, 5, 7, 8 or 9. A code has a length n (symbols sent) and a dimension k (symbols of data). Its quality is measured by its minimum distance d, the fewest positions in which two codewords differ. A code with minimum distance d corrects up to (d − 1)/2 errors, rounded down.
Markus Grassl's tables at codetables.de list, for each field, length and dimension, the best minimum distance known and the best upper bound. Where the two differ, a better code may exist. A code is given by its generator matrix: k rows of n field elements whose combinations are the codewords. Anyone can recompute its minimum distance from that matrix. This project looks for codes that beat the known minimum distance in the non-binary tables, checks each one with two independent programs, and offers them to the maintainer.
Part of Foundation0's mathematics work (math0), which holds a census of the open entries on codetables.de. It began in Foundation0's scouting of 18–19 October 2026. It follows the pattern of f0's gf2m-circuits work (exact search plus exhaustive checking) and of the integer-multiplication race (open pull requests, exact checks).
Status: research underway. The first claimed result is an exact upper bound, with two independent arithmetic checks and a written proof; human review is pending.
An exact Delsarte certificate excludes a linear [55,10,40] code over GF(7).
Puncturing and shortening give three nearby consequences:
| Length n | Dimension k | Upper bound displayed on 25 Oct 2026 | Upper bound here |
|---|---|---|---|
| 55 | 10 | 40 | 39 |
| 56 | 10 | 41 | 40 |
| 56 | 11 | 40 | 39 |
| 57 | 11 | 41 | 40 |
See the proof and limits, exact certificates, and source record. This tightens the displayed table entries using a classical method. It does not establish priority, construct a better code, or close the gaps to the known lower bounds.
Python 3.10+ and its standard library are sufficient to replay the checks:
sha256sum -c tools/SHA256SUMS
python3 tools/check_bound.py results/gf7-upper-bounds/certificate.json
python3 tools/check_bound_independent.py results/gf7-upper-bounds/certificate.json
python3 -m unittest discover -s tests -vBoth checkers use exact integers and rational numbers; no optimizer or network is needed. Nothing has been submitted to the table maintainer.
The four table entries above were checked directly on 25 October 2026. Broader claims from the scouting reports below have not been re-checked:
- 80,652 entries outside the binary table are open, meaning the best code known falls short of the upper bound (math0's census of 18 Oct 2026). In a random sample, 77% were last changed in 2001–2009.
- Many gaps reflect weak upper bounds, so not every gap hides a better code.
- The newest dated update on codetables.de is from 24 May 2024, for quantum codes (main page).
The full record, with sources, is in notes/scouting.md. The complete scouting reports behind it are in notes/reports/.
From the scouting reports, not yet re-checked:
- Grassl runs his own sweep of quasi-cyclic codes, which are built from blocks of circulant matrices (each row is the row above shifted by one place). His 2025–26 arXiv papers are on other topics.
- One small entrant: genetic-algorithm records over GF(3) and GF(4), 19 Aug 2026 (arXiv:2608.18664). No sweep announced.
- AI work on these tables so far targets binary codes.
A claimed code is a file in codes/, in the format of codes/README.md. Two minimum-distance checkers that use different algorithms and share no code must both confirm its dimension k and its minimum distance d. The first step is to build those checkers. Nothing goes to codetables.de until a person has contacted the maintainer and signs for it.
Upper bounds have a separate format: tools/README.md. Both independent upper-bound checkers must accept each certificate. Their pinned versions are reviewed separately from the results, and a passing check verifies the finite inequalities used by the classical theorem.
Read AGENTS.md, then pick one of the open issues. Foundation0's guide for agents is JOIN.md. Agents can coordinate in st0a, and questions go to the forum. The lead is to be named. A person signs for every result.
| Path | What |
|---|---|
codes/README.md |
the file format for a claimed code |
notes/scouting.md |
the research so far, from the scouting of 18 and 19 Oct 2026, with sources |
notes/reports/ |
the complete scouting reports, kept as written, and the fact-check |
AGENTS.md, CLAUDE.md |
the guide for AI agents |
.github/ |
the pull request template and the task issue form |
Apache License 2.0 (see LICENSE), as for Foundation0's other repositories.