Research / Constant-weight codes
Forty codewords raise the recorded lower bound on A(28,10,7) from 38 to 40.
Explicit construction, not peer reviewed- Question
- How many binary words of length 28 and weight 7 can there be if any two must differ in at least 10 positions?
- Result
- Forty binary words of length 28 and weight 7, any two differing in at least 10 positions. Brouwer’s table of constant-weight codes now lists 40 for this entry, credited to Warren Smith, up from 38.
- Boundary
- A lower bound only: the exact value is still open, between 40 and the table’s upper bound of 56. The construction has not been peer reviewed, and finding no 41 does not show that none exists.
- Who did what
- All search code and checks were AI-written, in OMEGA. Warren sent the result to the table’s maintainer. A listing is not a peer-reviewed publication.
- Evidence
- Brouwer’s tableThe 40 words
The explicit construction
Number the first 25 positions 5i + j, with i and j from 0 to 4, and let Z5 × Z5 shift i and j modulo 5. Positions 25, 26 and 27 stay fixed. The code is the union of the orbits of these four words, given by the positions of their ones:
{0, 1, 2, 6, 9, 14, 15}{0, 5, 10, 15, 20, 25, 26}{4, 8, 12, 16, 20, 25, 27}{2, 9, 11, 18, 20, 26, 27}
That gives 25 + 5 + 5 + 5 = 40 words. This site’s build rebuilds them from these four and compares the result with the listed file.
1110001001000011000000000000 0111010100100000100000000000 0011101010010000010000000000 1001100101001000001000000000 1100110010000100000100000000 0100000000011101010010000000 0000011100010010000110000000 0000001110101001000001000000 0010000000001110101001000000 0000000111010100100000100000 0001000000100110010100100000 1000001000000000111010100000 0100100001100000000011100000 0000010011001010010000010000 0000100000110011001000010000 0001000001000001100110010000 0100000100000000011101010000 1010010000010000000001110000 0000011001100100001000001000 1000000000111000100100001000 0000110000000001110001001000 1001000010000010000011001000 0010000010000001001100101000 0010100100000100000010011000 0101001000001000000000111000 1000010000100001000010000110 0100001000010000100001000110 0010000100001000010000100110 0001000010000100001000010110 0000100001000010000100001110 0000100010001000100010000101 1000000001000100010001000101 0100010000000010001000100101 0010001000100000000100010101 0001000100010001000000001101 0010000001010000001010000011 0001010000001000000101000011 0000101000000101000000100011 1000000100000010100000010011 0100000010100000010000001011
The checks
The result is independently checkable: anyone can test the 40 listed words directly. This site’s build recomputes them every time: 40 distinct words, length 28, weight 7, minimum distance 10. Programs written separately from the search passed the same test before the code was sent.
These programs have one AI author and rest on the same reading of the problem, so they are not independent human verification. No human review of them is on record, and the table’s listing does not state what its maintainer checked.
Attribution
The table is maintained by Andries Brouwer. Its entry reads Warren Smith, email 2026-11-02, shows A(28,10,7) ≥ 40
. It previously read 38, with an upper bound of 56, and the upper bound is unchanged. No upper-bound improvement is claimed.
What the searches did not find
- No direct extension, and no improving swap of up to 8 words (exhaustive, one implementation with a planted control).
- Across 701 candidate symmetry groups, the maximum was 40.
- Local search from this code stayed at 40 in 8 of 8 runs; from scratch it reached 35 in 8 of 8.
- A separate orbit-preserving search, 21 runs, reached at most 40.
- None of this bounds A(28,10,7) above: 41 may exist.
What is not claimed
Whether 40 appears in older sources is not settled: Handbook tables, paywalled journals and theses were not searched. The result is a table entry, not a peer-reviewed publication. A more general construction exists in draft, AI-checked only; it is not presented or claimed here.