All 51 with public certificates · Two van der Waerden rows compare against withdrawn preprint versions · Constructions of 23 to 30 September 2026 (UTC) · Earlier values read 24, 25 and 30 September 2026 · Not confirmed by any table maintainer
Table scrolls sideways for differences and sources.
| Object | Earlier value | Ours | Difference | Compared with |
|---|---|---|---|---|
| van der Waerden numbers, lower bounds | ||||
| W(4,6) | > 157,348 | > 255,606 | +98,258 | Monroe, arXiv 1603.03301v7 |
| W(5,6) | > 786,740 | > 1,278,030 | +491,290 | Monroe, arXiv 1603.03301 v4–v6 (a) |
| W(6,6) | > 1,555,549 | > 2,531,178 | +975,629 | Monroe, arXiv 1603.03301 v4–v6 (a) |
| W(10,3) | > 892 | > 937 | +45 | Komkov, arXiv 1701.05603v5 (b); > 937 added 13 Oct 2026, up from our > 935, see note and certificates |
| W(11,3) | > 1,187 | > 1,292 | +105 | Komkov, arXiv 1701.05603v5 (b) |
| Qubit stabilizer codes [[n,k]], minimum distance d | ||||
| [[17,5]] | d ≥ 4 | d = 5 | +1 | codetables.de, upper bound 5 |
| [[20,7]] | d ≥ 4 | d = 5 | +1 | codetables.de, upper bound 5 |
| [[21,8]] | d ≥ 4 | d = 5 | +1 | codetables.de, upper bound 5 |
| [[23,9]] | d ≥ 4 | d = 5 | +1 | codetables.de, upper bound 6 |
| [[43,32]] | d ≥ 3 | d = 4 | +1 | codetables.de, upper bound 4; pure; 3 exact checkers, all Claude-written (certificate, SHA-256 e7120cbb7e9e69dc…, 2 Oct 2026) |
| Covering arrays CAN(3,k,v), rows. Candidates (c) | ||||
| CAN(3,11,8) | 960 | 834 (c) | −126 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026; rechecked by an OpenAI GPT-6 agent with its own exact checker, 3 Oct 2026 |
| CAN(3,13,10) | 1,858 | 1,695 (c) | −163 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026 |
| CAN(3,13,11) | 2,064 | 1,991 (c) | −73 | Colbourn tables, Nov 2024 copy; construction array (folder covering-arrays), checked as listed under Covering arrays |
| CAN(3,15,12) | 3,486 | 2,891 (c) | −595 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026 |
| CAN(3,15,13) | 3,798 | 3,289 (c) | −509 | Colbourn tables, Nov 2024 copy; construction array (folder covering-arrays), checked as listed under Covering arrays |
| CAN(3,19,14) | ≤ 6,006 | 5,645 (c) | −361 | Colbourn tables, Nov 2024 copy (d); search arrays, Gemini-written exhaustive checker, 3 Oct 2026; rechecked by an OpenAI GPT-6 agent with its own exact checker, 3 Oct 2026 |
| CAN(3,19,15) | 6,415 | 6,085 (c) | −330 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026 |
| CAN(3,19,16) | 6,652 | 6,488 (c) | −164 | Colbourn tables, Nov 2024 copy; construction array (folder covering-arrays), checked as listed under Covering arrays |
| CAN(3,21,17) | 9,537 | 9,038 (c) | −499 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026; rechecked by an OpenAI GPT-6 agent with its own exact checker, 3 Oct 2026 |
| CAN(3,21,18) | 11,500 | 9,524 (c) | −1,976 | Colbourn tables, Nov 2024 copy; search arrays, Gemini-written exhaustive checker, 3 Oct 2026; rechecked by an OpenAI GPT-6 agent with its own exact checker, 3 Oct 2026 |
| Covering designs C(v,k,t), blocks | ||||
| 31 cells, listed below | 1,564 in total | 1,526 in total | −1 to −4 per cell | Covering Repository archive, 25 Sep 2026 (e) |
All 31 covering designs
| Object | Earlier value | Ours | Difference |
|---|---|---|---|
| C(36,22,4) | 19 | 18 | −1 |
| C(37,22,3) | 11 | 10 | −1 |
| C(41,7,2) | 43 | 42 | −1 |
| C(45,23,5) | 65 | 63 | −2 |
| C(48,8,2) | 46 | 45 | −1 |
| C(50,19,3) | 33 | 32 | −1 |
| C(53,19,4) | 127 | 123 | −4 |
| C(56,13,2) | 25 | 24 | −1 |
| C(56,24,3) | 24 | 23 | −1 |
| C(58,24,3) | 27 | 26 | −1 |
| C(62,17,2) | 19 | 18 | −1 |
| C(65,23,4) | 128 | 125 | −3 |
| C(70,21,2) | 16 | 15 | −1 |
| C(70,25,3) | 38 | 37 | −1 |
| C(73,25,3) | 40 | 39 | −1 |
| C(75,13,2) | 42 | 41 | −1 |
| C(79,24,3) | 62 | 61 | −1 |
| C(81,21,3) | 85 | 84 | −1 |
| C(82,19,2) | 25 | 24 | −1 |
| C(83,11,2) | 70 | 69 | −1 |
| C(86,20,2) | 25 | 24 | −1 |
| C(90,21,2) | 25 | 24 | −1 |
| C(91,24,3) | 85 | 84 | −1 |
| C(92,21,2) | 26 | 25 | −1 |
| C(92,24,3) | 85 | 84 | −1 |
| C(95,21,2) | 28 | 27 | −1 |
| C(95,25,3) | 85 | 84 | −1 |
| C(96,16,2) | 46 | 45 | −1 |
| C(96,17,2) | 41 | 40 | −1 |
| C(96,25,3) | 86 | 85 | −1 |
| C(97,25,3) | 87 | 85 | −2 |
Download irsi-certificates-2026-09-25.zip (48 KB, SHA-256 baba879c2874f0572ef7eae95f06ba9959c99a289706da0333d69563d781ca25). It holds the certificates for the 40 van der Waerden, code and covering-design entries and the three checkers shipped with them, short Python scripts written by Claude Opus agents that use only the standard library and check the full definition exhaustively. sh verify_all.sh checks all 40 and prints one VALID line per entry, in about 7 minutes on one core. The codes can also be compared with the live codetables.de entries, which still showed lower bound 4 on 30 September 2026. The ten covering arrays are in two further downloads: seven in irsi-ca-search-2026-10-03.zip and three (CAN(3,13,11), CAN(3,15,13) and CAN(3,19,16)) in folder covering-arrays of irsi-certificates-2026-10-01.zip. Each array passes check_ca.py from the second download and the Gemini-written verify_ca_gemini.py, for example python3 verify_ca_gemini.py ca_t3_k21_v18-recipeE-7001.txt --t 3 --k 21 --v 18 --n 9524.
How they were found
29 of the 31 covering designs came from an integer program over the point types of existing table entries, a computational form of known constructions; the other two follow from those by doubling or by deleting a point. W(4,6) and W(6,6) came from Xu’s product construction, with Heule’s composite cyclic moduli (226 and 2238) in place of the prime his theorem assumes, applied to Heule’s cyclic colourings and a 2-colour W(2,6) > 1131 colouring; the certificates are checked directly, so the bounds do not depend on that step. W(5,6) came from the Blankenship, Cummings and Taranchuk recurrence applied to our W(4,6). The others came from LLM-guided program search, agent-written searches, and, for five of the covering arrays, a closed-form construction that an agent wrote after studying the search output. We do not claim that any of these needed a language model rather than an exact method a person could have run.
Notes and limits
- (a) These values appear in v4 to v6 and were withdrawn in v7. The largest peer-reviewed values are > 540,025 and > 816,981 (Heule 2017).
- (b) The peer-reviewed values are > 889 and > 1,183 (Heule 2017). No maintained table exists for van der Waerden numbers; the earlier values are the largest found in literature searches on 24 September and 13 October 2026.
- (c) Colbourn’s covering array tables are no longer publicly available at their original location. We compared with Ulrike Grömping’s copy of their November 2024 state and did not screen later literature, so these ten may already be beaten. All ten arrays are public (see Check it).
- (d) The copy has no k = 19 row. 6,006 is its k = 22 value, which bounds k = 19 because columns can be deleted; the true earlier value may be lower.
- (e) Compared with the bulk archive exported by coveringrepository.com, the successor of the La Jolla Coverings Repository (April 2026 snapshot). On all 31 cells that archive equals the linked April 2026 snapshot. Per-cell pages, which can also show derived values, were read for only 2 of the 31 cells. C(96,16,2), C(96,25,3) and C(91,24,3) follow from other entries in the list.