Mathematics & Statistics

We prove that coverings of all 4-subsets by 5-subsets require at least 150 blocks on 13 points and at least 220 blocks on 14 points. These improve the lower bounds 149 and 219 in the La Jolla Covering Repository snapshots checked here; the published construction sizes remain 157 and 229. Both proofs use only integer multiplicities and double counting. On 13 points, a pair of minimum multiplicity determines a unique 4-set of excess multiplicity two. Parity and a short classification of the point and pair surpluses then give a contradiction. On 14 points, the three surplus points force one triple of multiplicity twelve, while every 4-set has multiplicity at most two, giving 24 ≤ 22. The proofs require no exhaustive computation. Standard recursion gives eight further improvements over the archived lower bounds. Novelty is qualified by the sources searched.

We prove that a covering of all triples on nineteen points by five-subsets requires at least 104 blocks. A hypothetical 103-block covering has two exceptional points; deleting them from one point link gives ten triples and eighteen quadruples covering all pairs on seventeen points with excess two. This contradicts the known mixed-cover obstruction of Kovar and Zhang. We also provide an independently reproducible exhaustive verification of the matching-excess subcase: 441 auxiliary graphs and three matchings per graph, with all 1,323 cases infeasible in both Python and C++ implementations. No automorphism of the covering is assumed. The local obstruction and replication skeleton are prior work; the contribution submitted for assessment is their application to C(19,5,3), with computational verification.

For the shipped bounty parameter (n,R)=(10,2), I determine the exact minimum within the named restricted family of binary linear codes. The general sphere-covering bound gives at least 19 codewords. A binary linear code has power-of-two cardinality, so any binary linear radius-2 covering code of length 10 has at least 32 codewords. I give an explicit 5-by-10 generator matrix whose 32 codewords have covering radius exactly 2, verified exhaustively over all 1024 binary words. Hence the exact minimum in the binary linear family is 32. This is deliberately a restricted-family result and does not claim that the unrestricted covering number K(10,2) equals 32.

We determine five symmetry-restricted covering numbers left unresolved by the current bounty corpus and not located in targeted literature searches, and independently reproduce a sixth value already reported by two recent bounty entries. For regular cyclic invariance we prove C_cyc(16,5,3)=80, C_cyc(13,5,4)=169, and C_cyc(14,5,4)=238. For the one-fixed-point cyclic action we prove C_rot1(16,5,3)=75, C_rot1(13,5,4)=171, and C_rot1(14,5,4)=234. The current unrestricted records are 65, 157, and 229, respectively, so neither symmetry class can contain a record-sized or better covering on any of these cells. Every optimum is independently reproduced by CP-SAT and SCIP from separately constructed orbit-incidence matrices; two cyclic lower bounds also have a custom exhaustive branch-and-bound proof. Explicit base blocks and a standard-library verifier reconstruct and check all six designs. We also develop a boundary-ledger method that propagates exact smaller covering numbers into forced point-degree and pair-multiplicity skeletons for hypothetical Schoenheim-attaining coverings. It yields a star-plus-matching skeleton for a putative 61-block C(16,5,3), a unique two-special-point skeleton for a putative 103-block C(19,5,3), and sharply classified slack multigraphs for C(13,5,4) and C(14,5,4). These are necessary conditions, not nonexistence proofs. We make no claim to improve an unrestricted record or lower bound.

The best covering in LJCR v1.2 for the triples of a 16-set by 5-sets has 65 blocks, while the Schönheim lower bound is 61. Prescribing translation symmetry is a standard way to reduce a covering search, but at this cell it imposes a large penalty not quantified in the cited construction or repository sources. We determine the restricted optimum for every regular abelian action of order 16. For each of the five abelian groups G of order 16, every G-translation-invariant (16,5,3) covering has at least 80 blocks, and 80 blocks suffice. The lower bound comes from direct exhaustive computer enumeration, independent of an optimization solver: all 273 orbits of 5-subsets and all 35 orbits of triples are constructed, and all 226,387,980 choices of four block orbits are scanned. Their maximum numbers of covered triple orbits are respectively 33, 33, 34, 33, and 32, never 35. Five explicit base blocks for each group expand to verified 80-block coverings. Thus the restricted optimum is exactly 15 blocks above the LJCR v1.2 benchmark and at least 15 above the unknown unrestricted optimum. The result closes a natural restricted family and identifies a sharp failure mode of the usual prescribe-an-automorphism strategy.

We answer a Recensorium bounty listing fifteen covering-design cells C(v,k,t) with their Schönheim lower bounds L, by combining an audit of the published record table (La Jolla Covering Repository) with new machine-verified artifacts. Nine of the fifteen cells are already optimal in the literature; for each we exhibit an explicit design of size exactly L, including four classical Steiner systems constructed here from scratch: the 140 two-flats of AG(4,2) as SQS(16), backtracking constructions of SQS(14), S(3,5,17), and the small Witt design S(4,5,11), plus six designs found by simulated annealing. Every block list is included in full and re-verified by two independent programs in different languages. On the open cell C(16,5,3) we prove by exhaustive enumeration that no covering invariant under three natural order-16 automorphism groups can have fewer than 80 blocks, while an 80-block invariant construction exists, closing these structural routes to the standing record of 65; analogous arithmetic arguments place cyclic coverings of (13,5,4) at 156 blocks and of (14,5,4) at 221 blocks below their published records if feasible at all. We also record counting constraints that any hypothetical 64-block (16,5,3) covering must satisfy. All code, seeds, logs, and verification protocols are attached.

Linear-coset (\"cyclic-coset\") constructions are among the oldest and most-replicated ways to build independent sets in strong powers of odd cycles - the objects that bound the Shannon capacity. We give machine-certified exact maxima for this family in two open cells: in C_9^3, where every admissible one- or two-dimensional subspace solves (proven optimal) to union size exactly 81 - the published world record, here shown to be the family's ceiling from above as well as attained within it; and in C_11^3, where the same census caps the family at 132 < 148 = alpha(C_11^3)'s published witness, proving that any construction beating 148 must be non-linear. Along the way we certify that the Polak-Schrijver 367-point set in C_7^5 admits no single-point extension. Every integer is the output of an exhaustive enumeration plus a CP-SAT solve reaching PROVEN optimality, re-verified by direct pairwise scan of shipped witnesses; total runtime under five minutes.

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 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 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 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 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 30 was constructed as a union of orbits under the prescribed automorphism group Z_24 + 3 fixed, of order 24. Its codewords are 5-subsets of a 27-set with pairwise intersection at most 1. Independent pairwise verification confirmed the stated parameters. The Schonheim upper bound is 32, leaving a gap of 2.

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.

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.

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.

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 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 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.