New lower bounds for binary constant-weight codes from prescribed-automorphism search, and two quantum codes meeting their upper bounds

Abstract

We give explicit binary constant-weight codes that raise the lower bounds on $A(n,6,w)$ in eight cells of Brouwer's table, all now listed there. The first round (9 October 2026) gave four values: $A(27,6,11)\ge 19305$, $A(27,6,12)\ge 24453$, $A(28,6,10)\ge 18642$ and $A(28,6,9)\ge 11154$, together with an explicit $A(24,6,12)$ code of 5566 words, now in the table's list of lost codes. The second round (14 October 2026) gave seven values, listed in the table since 15 October 2026: $A(29,6,10)\ge 27183$, $A(29,6,11)\ge 42042$, $A(29,6,9)\ge 14781$, $A(28,6,10)\ge 19110$, $A(27,6,12)\ge 24804$, $A(28,6,9)\ge 11388$ and $A(27,6,13)\ge 27783$, three of which replace first-round values. Every code is a union of orbits of the semilinear affine group $\mathrm{AΓL}(1,27)$, its index-2 subgroup, or $\mathrm{AGL}(1,23)$, acting on all but at most two coordinates. The orbits were chosen by maximum-weight independent set search on the orbit conflict graph: branch and bound in the first round, iterated local search in the second. Each code was checked by two separately written programs over all pairs of words and was regenerated from its generators and orbit representatives. We also report two qubit stabilizer codes, $[[20,11,4]]$ and $[[43,32,4]]$, which raise the lower bounds of 3 listed at codetables.de to the listed upper bound 4.