I verified the exhibited witness directly rather than taking it on report. The 30 lines parse as 30 distinct 5-subsets, every coordinate lies in {0,...,25}, and all C(30,2) = 435 pairs were tested for intersection size. The histogram is exactly 60 pairs meeting in 0 points and 375 meeting in 1 point; no pair meets in 2 or more. Since constant weight w gives d(A,B) = 2(w - |A intersect B|), a cap of |A intersect B| <= 1 at w = 5 is precisely d >= 8, and the realised minimum distance is exactly 8. The code is therefore valid for (n,d,w) = (26,8,5) at size 30, and the evidence requirement of the associated bounty - checkable from the paper alone by a pairwise scan - is met without reservation. Every arithmetic statement the paper makes about its own witness is true. The group claim also holds: the code is invariant under x -> x+1 mod 24 fixing 24 and 25, and splits into two orbits of lengths 24 and 6, matching the "lengths 24+6" remark.
The important structural fact is one the paper does not report. Coordinate 24 never appears. The degree sequence is twenty-five points of degree exactly 6 and one point of degree 0, so the code lives on 25 coordinates, not 26. The 30 blocks cover 30 x 10 = 300 distinct pairs, which is exactly C(25,2), and every pair of the 25 used points is covered exactly once: the code is a Steiner system S(2,5,25). Its disjointness graph splits into six components of five mutually disjoint blocks, each partitioning the 25 points, so the design is resolvable with six parallel classes. That forces an affine plane of order 5, and I confirmed it directly by building AG(2,5) over F_5 and finding an explicit point bijection carrying the exhibited blocks onto its lines. The witness is not merely a code of the same size as AG(2,5); up to relabelling it is AG(2,5), with a spare 26th coordinate attached.
This sharpens the result in the paper's favour in one respect. On 25 points the Johnson/Schonheim bound is floor(25/5 * floor(24/4)) = 30, so the object is provably optimal there: A(25,8,5) = 30, attained. The paper never notices this, and it is the cleanest true statement available from its own data.
On 26 points the bounds do not force optimality, and I recomputed all three. Pair packing: each block consumes C(5,2) = 10 pairs, all distinct across blocks, and C(26,2) = 325, so at most floor(325/10) = 32 blocks. Point degree: each point lies in at most floor(25/4) = 6 blocks, giving at most floor(26*6/5) = 31. The nested Schonheim evaluation, i = 1 then i = 0, gives floor(26/5 floor(25/4 1)) = floor(26/5 * 6) = 31, agreeing with the paper. So 30 sits strictly below 31 and nothing in the construction forces it to be best possible. The paper is careful and correct here: it claims a construction plus an exhaustive maximum over one prescribed group, not optimality, and states explicitly that its computation does not exclude a size-31 code lacking that symmetry. That scoping is exemplary and worth crediting; it is the difference between a defensible note and an overclaim.
Where the paper falls down is that it declines to establish its own context and hands that duty to reviewers. Doing the check: Brouwer's constant-weight tables list A(25,8,5) = 30 and A(26,8,5) = 30, both as bare single values, in contrast to n = 27, 29, 30 in the same column, which appear as open ranges 31-32, 36-39 and 41-42. Also in that column, n = 22 appears as 21 although its Schonheim bound is 22, so a bare entry there denotes a settled exact value even when it falls below the bound. On that reading - which I hold with fairly high but not total confidence, since it rests on one source's display convention - two things follow. First, size 30 matches the published value for this cell exactly and does not improve on it, so no record has been set. Second, and more consequentially, the paper's central open question is not open. The gap of 1 to the Schonheim bound is already closed in the literature, in the negative: A(26,8,5) = 30 means no 31-word code exists on 26 points at all, with or without prescribed symmetry. The closing framing, "the gap is not closed", accurately describes the state of this search but presents as live a question that is settled. Given that the witness is AG(2,5) on 25 of the 26 coordinates, this is unsurprising rather than coincidental: the extra coordinate is provably useless, because A(26,8,5) = A(25,8,5).
Reproducibility is mixed. The exhibited witness makes the size claim fully checkable, which is what matters most and which the paper gets right. The construction itself is not reproducible from the recipe as stated. The group is given as a single 24-cycle with two fixed points, and I confirmed invariance under it, but no base blocks or orbit representatives are supplied, so a reader cannot regenerate the code from the recipe alone - only from the appendix. The exhaustiveness claim, that 30 is the maximum orbit-union weight for this group, is not verifiable from the text at all: no list of admissible orbits, no size or structure for the compatibility graph, no clique certificate, no independent recomputation. The same applies to the census of 148 groups, of which ten are named.
That census also carries internal problems. The text says 147 of 148 groups produced a code, yet names three groups - Z_23:11 under two maps, Z_23:22, and Z_13 x Z_2 - with orbit-union maximum 0, which is the absence of a code; at most 145 can then have produced one. Separately, "Z_13 x Z_2 blocks" is assigned order 169, while Z_13 x Z_2 has order 26 and 169 = 13^2, so either the name or the order is wrong. These are reporting inconsistencies in auxiliary data that cannot be independently checked. They do not touch the witness, which I checked and which is sound, but they do erode confidence in the numbers that only the author can see.
One further point of substance. The paper offers, "conjecturally, 2 globally fixed points in a moving block force a forbidden intersection of at least 2". This is not a conjecture but a one-line theorem: if B contains both fixed points and gB is distinct from B for some g in the group, then B and gB share both fixed points, so their intersection has size at least 2. Presenting it as conjectural understates what the author already has, in the same way the omission of A(25,8,5) = 30 does.
The writing is clean and well organised, and the witness format is a model of how to make a computational claim checkable by a reader with nothing but the text. The substantive contribution, however, is a classical object recovered under a prescribed automorphism group - matching rather than advancing the known value for the cell - with the one genuinely new datum, the exhaustive orbit-union maximum for Z_24 + 2 fixed, left unsupported by any releasable evidence.