A constant-weight code is an error-correcting code in which every codeword has the same Hamming
weight. For binary codes, this means that each codeword
contains the same number 
of ones. Let 
be the greatest number of binary codewords
of length 
, weight 
, and pairwise Hamming
distance at least 
(Dzoba 2026).
For pairwise Hamming distance at least 4, 
equals the independence
number of the Johnson graph 
. Two distinct binary codewords
of weight 
are adjacent in this graph precisely when their Hamming
distance is 2.
Dzoba (2026) constructed codes giving the following improved lower bounds. These are existence bounds, not claims of optimality.
 |  | lower bound for  |
| 27 | 11 | 19305 |
| 27 | 12 | 24804 |
| 27 | 13 | 27783 |
| 28 | 9 | 11388 |
| 28 | 10 | 19110 |
| 29 | 9 | 14781 |
| 29 | 10 | 27183 |
| 29 | 11 | 42042 |
Each code is a union of orbits of a prescribed permutation group. Dzoba (2026) reports that Claude designed and ran the searches and produced the constructions under the author's direction. Claude and Codex supplied separately written checking programs.
Chang (2026) obtained the upper bound
 |
The proof supplies a certificate checked using exact rational arithmetic for a semidefinite programming bound less than 674. Chang (2026) reports using GPT and Claude to prepare the computational code.
." 30 Oct 2026.