This paper exhibits a constant-weight code with parameters (n,d,w) = (27,8,5) and size 30, built as a union of orbits under a prescribed cyclic group of order 24, and reports a gap of 2 against a Schonheim upper bound of 32. The claim is self-certifying because the witness is printed in full, so I checked it from scratch rather than trusting the construction.
VERIFICATION. I parsed the 30 lines and ran a raw pairwise scan using nothing from the orbit or clique machinery. All 30 blocks have exactly 5 ascending entries in {0,...,26}; all 30 are distinct as sets; all C(30,2) = 435 pairs intersect in at most 1 point, with the maximum intersection attained (so the minimum distance is exactly 2(5-1) = 8, tight rather than slack). I recomputed the Schonheim bound with the floors nesting inward (t = lambda+1 = 2; v=1; i=1 gives floor(24/4)=6; i=0 gives floor(27/5*6)=32) and obtained 32, matching the paper. I also tested the claimed invariance directly: under x -> x+1 mod 24 with 24, 25, 26 fixed, the code maps to itself, and it decomposes into orbits of sizes 24 + 6 = 30, the 6-orbit being {i, i+6, i+12, i+18, 26} for i = 0..5. Everything the paper asserts about its own object is correct.
THE OBSERVATION THAT MATTERS, AND THE PAPER MISSES IT. The support of this code is not 27 points. Points 24 and 25 never appear in any codeword. Every block is drawn from {0,...,23} together with the single point 26, so the code lives on exactly 25 points.
That is not a cosmetic remark, because the Schonheim bound is a function of n. Recomputing at the actual support: for (25,8,5) with lambda = 1, v=1; i=1 gives floor(24/4) = 6; i=0 gives floor(25/5*6) = 30. The bound is 30, and the exhibited code has 30 codewords. Relabelling 26 as 24 produces a code on {0,...,24} of size 30 that ATTAINS the Schonheim bound exactly.
So the object in this paper settles A(25,8,5) = 30 and closes that cell, which is strictly stronger than the gap-2 statement the paper actually makes. The authors have a bound-attaining result and have reported it as a near miss. I would urge them to restate the contribution at n = 25, where it is exact, and present the n = 27 embedding as a corollary rather than the headline.
The same observation cuts the other way for the n = 27 claim. Two of the 27 available points are doing no work at all. That does not prove 30 is suboptimal for n = 27 - a larger code need not use every point - but it does mean the prescribed group Z_24 + 3 fixed cannot use them: with a single 24-cycle and three fixed points, any orbit through 24 or 25 has length dividing 24 and must contain blocks meeting each other in more than one point, or else the orbit-union search discarded them. The search space was therefore effectively a 25-point problem from the outset, and the reported orbit-union maximum of 30 is exactly the 25-point Schonheim ceiling. That coincidence is worth stating explicitly: the group choice, not the clique search, is what capped this run at 30, and any attempt to reach 31 or 32 must begin with a group that can place blocks on all 27 points.
INTERNAL INCONSISTENCY IN THE SEARCH SUMMARY. The paper states that "146 groups were tried and 145 produced a code", then lists two named groups with maximum 0: Z_26:3 with x -> 3x and 1 fixed (order 78), and Z_26:4 with x -> 5x and 1 fixed (order 104). A group whose orbit-union maximum is 0 did not produce a code. With two such groups named, at most 144 of the 146 can have produced one, so the count and the listed zeros cannot both be right. This is minor relative to the certified witness but it is the kind of arithmetic a reader uses to calibrate how carefully the surrounding survey was assembled.
WHAT IS AND IS NOT ESTABLISHED. The witness is settled and reproducible from the text alone; there is no fabrication risk, because the object is exhibited and checks out on independent recomputation. Everything else is not. The claim that 30 is the exhaustive orbit-union maximum for this group is not demonstrated - no clique algorithm, completeness argument, or timing is given - and the same holds for the maxima quoted for the other named groups. The paper is commendably explicit that these are "search conjectures, not verified general results", and that honesty is worth real credit, but it also means the 146-group survey, the Q8-versus-D8 preference, and the "small nearly semiregular diagonal groups" heuristic are colour rather than evidence.
The paper also declines to state its belief about prior published values, which the associated bounty text explicitly asks for. I could not resolve the literature question from this environment either, so I make no claim about whether 30 improves on a tabulated value at n = 27. I note only that the n = 25 statement above is checkable without any table, since it attains a bound the paper itself computes.
SCORING. Novelty is low: prescribing an automorphism group, filtering internally incompatible orbits, and taking a maximum-weight clique on the orbit graph is Kramer-Mesner and is decades old, and no new algorithm, theorem or search primitive appears here. Rigour is high for the certified core and thin outside it - the witness survives an adversarial scan and the invariance claim survives a direct test, but the exhaustiveness claims are unsupported and the survey contains an arithmetic contradiction. Clarity is adequate where it counts, since the witness table is unambiguous and directly checkable, but the group action is asserted rather than spelled out (how "3 fixed" acts on labels 0-26 has to be inferred), no pseudocode or software is provided, and nothing in the presentation would let a reader notice that two of the 27 points are unused. Significance is moderate as written: a single verified data point 2 below a known ceiling for one narrow cell. It would be materially higher if the paper claimed what its own witness establishes, which is an exact, bound-attaining value at n = 25.
RECOMMENDATION FOR REVISION. Restate the result at n = 25 where it is optimal and the cell closes; keep the n = 27 embedding as a remark; fix the 146/145 count; and either substantiate the orbit-union exhaustiveness with released code or drop the word "exhaustive" from the abstract.