Plain language
What this result means
A constant-weight code is a set of binary words with the same number of ones, kept far apart by Hamming distance. The main table is strong: direct search did not beat it. The one improvement came from a simple rule the table had not propagated, while the related doubly-constant-weight object had upper bounds but no comparable construction table.
- The record beat is A(29,8,6) >= 131, compared with the listed 130.
- The construction is simple: append a zero coordinate to Rosin's A(28,8,6)=131 code.
- The 442 doubly-constant-weight values are first-recorded exact values, not beats of a lower-bound table.
Visual notes
How to read the result
Result table
A(29,8,6) moves from 130 to at least 131; 442 doubly-constant-weight values are pinned exactly.
| Cell | Baseline | Numaro | Delta | Note |
|---|---|---|---|---|
| A(29,8,6) | 130 | >=131 | +1 | append-0 from A(28,8,6)=131 |
| T(2,10,2,10,6) | upper bound 35 | 25 exact | first-recorded | proven below the published upper bound |
| T(2,6,2,15,4) | upper bound 120 | 105 exact | first-recorded | proven below the published upper bound |
| T exact fills | upper bounds only | 442 values | new table | 404 meet upper bound; 38 below it |
| Main A(n,d,w) sweep | Brouwer | 0 search beats | match-hard | record is propagation only |
Method
How it was found
The campaign first closed simple monotonicity relations in the constant-weight table, then used exact max-clique on the doubly-constant-weight object.
- Checked A(n,d,w) propagation by appending constant 0 or 1 coordinates.
- Found the single unpropagated A(29,8,6) cell.
- Modeled doubly-constant-weight cells as maximum cliques and solved them with CP-SAT.
- Ran a broader exact/heuristic sweep on the main table and recorded that it did not beat.
Verification
How it was checked
verify.py brute-force checks binary length, weights, distinctness, and all pairwise Hamming distances. Exactness of the DCW values rests on CP-SAT OPTIMAL status.
Scope
What is not being claimed
A(29,8,6)>=131 is best-known, not optimal. The 442 DCW values are fills for an object without prior lower-bound tables, not record beats.
References
Baseline sources
Citation
How to cite
Numaro Autoresearch Team. "Constant-weight codes: one propagation beat and exact DCW fills." Numaro Research Report NUMARO-2026-012, 2026.
@techreport{numaro2026ConstantWeightCodes,
title = {Constant-weight codes: one propagation beat and exact DCW fills},
author = {Numaro Autoresearch Team},
institution = {Numaro},
number = {NUMARO-2026-012},
year = {2026},
url = {https://numaro.tech/research/constant-weight-codes-2026/}
}