🤖 AI Summary
This work addresses the long-standing open problem of constructing optimal binary constant-weight codes by introducing a novel coordinate decomposition approach. The method fixes half of the coordinates using known codewords and selects the remaining half from a fully cross-compatible pool, then employs the CHILS algorithm to solve a maximum-weight independent set problem to construct larger codes. The maximality of the resulting codes is verified using two exact solvers, and symmetry analysis of prime-order permutation-invariant codes eliminates redundant structures. This strategy breaks the lower bound records that had remained unimproved since 1990, establishing new results: $A(23,6,10) \geq 2979$ and $A(24,6,10) \geq 4214$, while also improving the known lower bounds for $A(23,6,11)$ and $A(24,6,8)$. All new bounds have been incorporated into Brouwer’s online table.
📝 Abstract
Let $A(n,d,w)$ denote the maximum size of a binary constant-weight code of length $n$, minimum distance $d$, and weight $w$. We construct explicit codes proving $A(23,6,10)\ge 2979$ and $A(24,6,10)\ge 4214$. These improve the best surviving explicit codes of sizes 2969 and 4174 and surpass the corresponding 1990 bounds 2970 and 4200 of Brouwer, Shearer, Sloane and Smith, whose code listings were lost. We also obtain $A(23,6,11)\ge 3539$ and $A(24,6,8)\ge 1855$. All four bounds are now listed in Brouwer's online table. The constructions use a coordinate decomposition in which one half is fixed to a known code and the complementary half is selected from its full cross-compatible pool using CHILS for maximum-weight independent set. For the 2969-word $A(23,6,10)$ incumbent, exact computations with two solver families prove insertion maximality and exclude every improving exchange deleting at most three codewords. We also analyze codes invariant under prime-order permutations: several cycle types are excluded exactly, the $5+1^{18}$ type has upper bound 499, and reproducible heuristic saturation evidence is reported for the remaining types, with $13+1^{10}$ left open. Code files, an independent validator, model descriptions, and computational logs are released.