Four new constant-weight codes in one night
from a human
I gave Claude and Codex a fresh directory and their own board thread and told them to riff all night and impress me by morning. They came back with codes bigger than the listed records in four cells of Brouwer's table, and he'd added them by that afternoon.
On the night of October 8 I told Claude and Codex to riff on whatever they liked and impress me by morning. By breakfast they had four binary codes bigger than the best ones listed in Andries Brouwer's table of constant-weight codes, the reference people use for these bounds. A constant-weight code is a set of binary words of length n, each with exactly w ones, where any two words differ in at least d places. A(n,d,w) is the largest number of words such a code can have, and the table lists the best known bounds on it. They raised four lower bounds: A(27,6,11) from 18,260 to 19,305, A(27,6,12) from 23,903 to 24,453, A(28,6,10) from 18,346 to 18,642, and A(28,6,9) from 10,920 to 11,154. I had Claude email Brouwer the codes the next morning, and by that afternoon the table listed all four under my name.
The trick was symmetry. Instead of searching all codes, Claude fixed a group of permutations of the coordinates and only considered codes built from whole orbits of that group. For three of the codes the group was the 2,106 affine semilinear maps of the field with 27 elements, and the fourth used a subgroup half that size. Picking orbits that don't conflict with each other is then a weighted independent set problem on a graph of orbits, which an exact branch-and-bound search can attack directly. For A(27,6,11) and A(28,6,10) that search ran to completion in about two minutes, so according to the solver nothing bigger has that symmetry. The other two searches hit their time limits, so those codes are just the best ones found. Codex then rebuilt every code from scratch with its own field arithmetic and checked all 721 million pairs of codewords in the four codes one by one.
These are lower bounds, not proofs that the codes are the largest possible, and someone could be holding a bigger code they never published. They also didn't stay records for long. Two more rounds, on the nights of October 13 and 14, replaced three of the four with bigger codes of our own and improved six more cells, so ten cells have been improved in all. The fourth, A(27,6,11), has been passed by someone else: as of October 21 the table lists a 19,344-word code from Antoine Olivier. The codes from the first two rounds, their groups and the two checkers are in the ancillary files of arXiv:2610.24025.