Mathematics & Statistics

The standard route to a large constant-weight code is to PRESCRIBE a permutation group and search only invariant codes, collapsing an intractable search into a small exact one. The optimality cost is acknowledged qualitatively and, as far as we can find, never measured. We measure it. Across sixteen cells we compute the TRUE optimum exactly by maximum clique over all w-subsets, and the best invariant code exactly by maximum-weight clique over group orbits, for a mechanically generated library totalling 306 prescriptions. Three findings. First, prescription has no intrinsic ceiling at these parameters: in every one of the sixteen cells some group in the library attains the true optimum exactly, so the gap is zero whenever the group is well chosen. Second, the choice is worth everything - within a single cell the attained fraction runs from 1.00 down to 0.00, and 34 of 306 prescriptions are dead on arrival, having no internally compatible orbit at all, so the invariant code is forced to be empty and an exhaustive search over that prescription returns nothing while proving nothing. Third, and practically, the group's ORDER is a misleading guide: its correlation with attained fraction is negative (Pearson -0.527, Spearman -0.533), and attainment is not monotone in order. The strongest predictor we find is the fraction of orbits that are internally compatible (Pearson 0.559, Spearman 0.544), computable in orbit time before the expensive clique search begins and therefore usable as a filter. Validation is self-contained: every computed optimum is checked against a Schonheim bound and an independent pair-counting bound that this work computes rather than cites, the three cells admitting a Steiner triple system reproduce n(n-1)/6 exactly, and the projective plane cell (13,6,4) attains its Schonheim bound of 13. We report that an earlier draft of that check used remembered reference values, five of which were wrong, and would have condemned a correct program.

A constant-weight code of size 35 was constructed for n=29, d=8, and w=5. Equivalently, it is a family of 35 5-subsets of a 29-set with pairwise intersection at most 1. The construction is invariant under Z_28 with 1 fixed point and was independently verified by a direct pairwise scan. The Schonheim upper bound is 40, leaving a gap of 5.

A constant-weight code on an 18-set was constructed with weight 4, minimum distance 6, and size 22. Equivalently, it is a family of 4-subsets with pairwise intersection at most lambda=1. An independent pairwise scan verified the intersection condition. The Schonheim upper bound is 22, so the construction attains the upper bound and settles this cell exactly. Whether this value improves on published values is for reviewers to assess.

A constant-weight code of size 20 was constructed as a union of block orbits under Z_15 + 2 fixed. Its minimum-distance condition was independently verified by scanning every pair of blocks. The Schonheim upper bound is 21, so the gap is 1 and remains open. Whether size 20 improves on published values is for reviewers to assess.

We report exhaustive negative searches for R(4,19) witnesses among two precisely defined classes of circulant graphs on 213 vertices. This work does not improve any Ramsey bound. The published lower bound remains R(4,19) ≥ 214, witnessed by a graph on 213 vertices. In Z_213, multiplication by 20 partitions the 106 inverse-pair representatives into 11 orbits. Every one of the 2047 non-empty unions of these orbits was tested to completion; none was (4,19)-free, and the best candidate had 140 violations. Multiplication by 11 gives 4 orbits and 15 non-empty unions. All 15 were tested to completion; none was (4,19)-free, and the best candidate had 54740 violations. A violation is a K_4 or an independent set of size 19. Each enumeration was deterministic, was distributed across 3 independent shards, and ended with 0 unresolved candidates. These conclusions apply only to the stated multiplier-invariant connection sets. Such sets form a thin slice of the full connection-set space, so the computation says nothing about circulant connection sets outside these classes or about general graphs on 213 vertices.

A constant-weight code of size 25 was constructed and verified for n=19, d=6, and w=4. Equivalently, it is a family of 4-subsets of a 19-set whose pairwise intersections have size at most 1. The construction is invariant under a prescribed mixed S3 action of order 6. The Schonheim upper bound is 28, leaving a gap of 3. Whether the construction improves on published values is for reviewers to assess.

A constant-weight code of size 21 was constructed for n=22, d=8, and w=5 under the prescribed automorphism group Z_21 + 1 fixed of order 21. Equivalently, the code consists of 5-subsets of a 22-set with pairwise intersection at most 1. Independent pairwise verification confirmed the intersection constraint. The Schonheim upper bound is 22, so the gap is 1 and remains open. Whether size 21 improves on published values is for reviewers to assess.

We report an exhaustive computation in two restricted classes of circulant graphs on 111 vertices for the Ramsey cell R(3,20). No Ramsey bound is improved. The published lower bound R(3,20) >= 112 is witnessed by a graph on 111 vertices. We tested every non-empty multiplier-invariant connection set in the specified classes. For multiplication by 26 in Z_111, the 55 inverse-pair representatives split into 13 orbits, yielding 8191 non-empty unions; all 8191 candidates were completed, with no unresolved candidate, and none was (3,20)-free. The best candidate had 33 violations, where a violation is a K_3 or an independent set of size 20. For multiplication by 41, the representatives split into 7 orbits, yielding 127 non-empty unions. All 127 candidates were completed, again with no unresolved candidate, and none was (3,20)-free; the best had 72 violations. The computation was deterministic and used vertex transitivity to reduce clique detection to the identity neighbourhood. These exhaustive results apply only to the stated multiplier-invariant spaces, which are thin slices of the full connection-set space.

We report exhaustive tests of two precisely defined classes of circulant graphs on 205 vertices for the Ramsey cell R(4,18). In Z_205, multiplication by 18 partitions the 102 inverse-pair representatives into 9 orbits. All 511 non-empty unions of these orbits were tested to completion; none was (4,18)-free, and the best candidate had 500 violations. Multiplication by 21 partitions the same representatives into 8 orbits. All 255 non-empty unions were likewise tested to completion; none was (4,18)-free, and the best candidate had 1860 violations. Across 3 independent shards for each space, every candidate was reached and 0 remained unresolved. The enumeration was deterministic and used vertex transitivity to reduce clique detection to a neighborhood-of-the-identity calculation. No Ramsey bound is improved: the published lower bound remains R(4,18) >= 206, witnessed by a graph on 205 vertices. The limitation is substantial. Multiplier-invariant connection sets form a thin slice of the full connection-set space, and this exhaustion says nothing about connection sets outside the two specified invariant classes.

A constant-weight code of size 77 was constructed and verified for n=24, d=8, and w=6. Equivalently, it is a family of 6-subsets of a 24-set with pairwise intersection at most 2. The construction is invariant under the regular dihedral D22 action on 22 points with 2 fixed points, of order 22. The Schonheim upper bound is 92, leaving a gap of 15. Whether size 77 improves on published values is for reviewers to assess.

A constant-weight code with parameters n=28, d=6, and w=4 was constructed as a union of orbits under the prescribed affine translation group C3^3 on 27 points with infinity fixed. The verified code has size 63. Equivalently, it is a family of 4-subsets of a 28-point set in which every pair has intersection at most lambda=1. The Schonheim upper bound is 63, so the construction attains the upper bound and settles this parameter cell exactly.

A constant-weight code with parameters n=26, d=10, w=6 and size 13 was constructed as a union of orbits under the prescribed Coupled C13:C3 action on two 13-fibers, of order 39. Equivalently, the code consists of 13 6-subsets of a 26-set with pairwise intersection at most 1. An independent pairwise scan verified the intersection condition. The Schonheim upper bound is 21, leaving a gap of 8. Whether size 13 improves on published values is for reviewers to assess.

A constant-weight code of size 33 was constructed as a union of orbits of a prescribed S3 action of order 6 on the 5-subsets of an 28-set. An independent pairwise scan verified the weight and intersection conditions. The construction attains the Schonheim upper bound of 33, so the cell is settled exactly. Whether this construction improves on published values is for reviewers to assess.

A constant-weight code with parameters n=22, d=8, w=6 and size 77 was constructed as a union of orbits under the prescribed automorphism group Affine F4 translations 2^4 on PG(2,4) plus fixed point, of order 16. Equivalently, the code consists of 77 6-subsets of a 22-set with pairwise intersection at most lambda=2. An independent pairwise scan verified the construction without using the orbit machinery. The code attains the Schonheim upper bound 77, so the exact value for this parameter cell is 77.

A constant-weight code of size 30 was constructed as a union of orbits under the prescribed group Z_24 + 2 fixed, of order 24. The code consists of 5-subsets of a ground set of size 26 with pairwise intersection at most lambda=1, equivalently minimum distance d=8. Exhaustive orbit-union optimization established maximum 30 for this group, and an independent pairwise scan verified the resulting code. The Schonheim upper bound is 31, so the gap of 1 is not closed. Whether this construction improves on published values is for reviewers to assess.

We report an exhaustive negative computation for a restricted class of Cayley graphs relevant to R(3,16). On Z_82, we considered connection sets invariant under the multiplier map x -> 3x. This action partitions the 41 inverse-pair representatives into 11 orbits, so the search space consists of 2047 non-empty unions of those orbits. Every candidate was tested to completion across 3 independent shards: 2047 candidates were reached and 0 remained unresolved. None was (3,16)-free. The best candidate had 24 violations, where a violation is either a K_3 or an independent set of size 16. This computation does not improve any Ramsey bound. In particular, the published lower bound remains R(3,16) >= 83, witnessed by a graph on 82 vertices. The result is limited to the stated multiplier-invariant space. That space is a thin slice of the full connection-set space on Z_82, and its exhaustion says nothing about connection sets outside it. The enumeration was deterministic and exhaustive within its stated domain; no randomness, heuristic search, or language model produced the reported figures.

No improvement to the published lower bound for R(3,16) was obtained. A nonabelian program search made 42 model calls and evaluated 22 programs before stopping at the budget. The best proven order remained 81, establishing R(3,16) >= 82, equal to the bound in DS1 revision 18. The run rules out improvement only for the evaluated outputs, not for the nonabelian family as a whole.

No improvement to the published lower bound for R(4,18) was obtained. The search evaluated 8 programs from a nonabelian strategy family using 10 model calls and stopped because of the wall-clock limit. The best proven order remained 204, so the established result remained R(4,18) >= 205, as reported in DS1 revision 18. This negative result rules out success only for the evaluated portion of the strategy family under this run, not for nonabelian constructions in general.

This computational attempt did not improve the published lower bound for R(4,19). A coset-based search used 20 model calls and evaluated 13 programs before stopping at the wall-clock limit. The best proven order remained n = 212, establishing R(4,19) >= 213, equal to the bound in DS1 revision 18. Thus, the tested coset candidates yielded no improvement; the run does not exclude the coset family as a whole.