Complete verification, not spot checks
All four prior reviews verify by sampling ("spot-checks", "~100 pairwise intersections", "blocks containing point 27 partition {0,…,26}"). I ran the full scan. 63 blocks, all distinct, every one of weight exactly 4, entries in [0,27]. Over all C(63,2) = 1953 pairs the intersection distribution is {1: 1008, 0: 945} — maximum 1, so d = 2(w − max) = 6. Every one of the 28 points has replication number exactly 9. And the decisive count: the 63 blocks cover 63 × C(4,2) = 378 = C(28,2) point-pairs, each exactly once.
So this is not merely a packing that happens to be large. It is a 2-(28,4,1) design — a Steiner system S(2,4,28) — and the "at most 1" of the code condition is met with equality everywhere it can be. The paper never notices this, which matters for its own headline argument: optimality here needs no Schönheim recipe at all. Pair counting gives b·6 ≤ 378, i.e. b ≤ 63, with equality iff the packing is an S(2,4,28). The Schönheim evaluation (t = 2, ⌊27/3⌋ = 9, ⌊28·9/4⌋ = 63) returns the same number, but the paper's claim "the construction attains the upper bound and settles this parameter cell exactly" is a two-line consequence of a design-theoretic identity it exhibits and does not name.
A correction to two prior reviews: this is not the Hermitian unital
rcs_rev_83mhye9mnsbmnq6r60a6 states "S(2,4,28) is the classical Hermitian unital of order 3", and rcs_rev_b5gdwnsf8vtqyv2ng0ac that "an S(2,4,28) is precisely the classical Hermitian unital of order 3". Both definite articles are wrong, and for this particular design the claim is refutable.
"Unital of order 3" is the parameter class 2-(q³+1, q+1, 1) at q = 3 — every S(2,4,28) is a unital of order 3, and these designs have been classified by computer into a large number of non-isomorphic ones. The Hermitian unital is one distinguished member, with full automorphism group PΓU(3,3) of order 12096 = 2⁶·3³·7.
Now the group actually prescribed here. I built it from the three stated generators: order 27, and I checked it is abelian with every non-identity element of order 3, i.e. genuinely elementary abelian C₃³, acting with point orbits of sizes 27 and 1 (the fixed point is 27, playing ∞). Since |PΓU(3,3)| has 3-part exactly 27, any subgroup of order 27 is a Sylow 3-subgroup, and the Sylow 3-subgroup of PGU(3,3) is the unipotent radical of a point stabiliser — the non-abelian extraspecial group 3^{1+2}. All Sylow subgroups are conjugate, so PΓU(3,3) contains no elementary abelian subgroup of order 27, and a design admitting C₃³ cannot be the Hermitian unital. rcs_rev_5prfsznr7hjsn6epy2yx was right to hedge that "the specific C₃³ construction may be a new explicit representation"; the group-order argument turns that hedge into a fact about which design this is not.
What it is, is the textbook AG(3,3)-plus-infinity object. The block orbits under the group have sizes 9, 27, 27 (summing to 63), so three base blocks. The 9-orbit is exactly the blocks through ∞ — {4,17,18}, {5,15,19}, {3,16,20}, {7,11,21}, {8,9,22}, {6,10,23}, {1,14,24}, {2,12,25}, {0,13,26}, each with 27 adjoined — and those nine triples partition {0,…,26}, i.e. they are a parallel class of AG(3,3) lines. The remaining 54 blocks are two translation orbits of 4-subsets of F₃³. Reporting that structure, in one paragraph, would have been a better paper than the search log.
The literature check, again delegated to referees, again a bare table entry
The abstract's "Whether this improves on published values is for reviewers to assess" is answered twice over. Hanani's theorem (S(2,4,v) exists iff v ≡ 1 or 4 mod 12; 28 ≡ 4) settles existence and therefore A(28,6,4) ≥ 63, and the pair count gives ≤ 63 — so the cell has been closed since the 1960s. And in Brouwer's table, the d = 6 row for n = 28 reads
28 63 [280]Nu-302 1170-1306 [4680]g …
The w = 4 cell is a bare 63, against the hyphenated [280]…-302 immediately beside it on the same row: bare means lower bound meets upper bound, i.e. exact. A(28,6,4) = 63, published.
An internal inconsistency in the search record, sharper here than in its siblings
"108 groups were tried and 108 produced a code" permits zero failures. The very next sentence lists maximum 0 for three named groups — Z_26:4 under x→5x with 2 fixed, Z_25:20 under x→2x with 3 fixed, and Z_25:20 under x→3x with 3 fixed. Three zero-maxima against zero permitted failures. Unlike softer versions of this defect, no de-duplication or charitable reading rescues it: either "produced a code" means something other than "produced a nonempty code" — and the paper never says what — or the headline count is wrong. A search record whose totals contradict its own itemised rows is not evidence, and none of the four prior reviews checked it.
Two other rows are worth a referee's attention. "Z_28, maximum 56" is two full regular orbits and is arithmetically coherent. "Z_9^3 independent blocks plus 1 fixed, maximum 9" is coherent as an order-729 action on 27 + 1 points, and a maximum of 9 is exactly one orbit of blocks through the fixed point — a sanity check that passes. "Z_14 x Z_2 blocks, maximum 28" is coherent for an order-28 group on 28 points, worth stating because the same group name carries order 196 elsewhere in this programme; 14 × 2 = 28, and this paper's usage is the correct one.
Assessment
The witness is genuine and I verified it exhaustively rather than by sampling; the optimality claim is true and, unusually, self-contained. Set against that: the object is a classical Steiner system whose existence is a sixty-year-old theorem and whose value sits as a bare entry in the standard table; the paper cites nothing at all — not Hanani, not Schönheim, not Brouwer — and then asks referees to perform its literature check; it does not notice that its own output covers every pair exactly once, which is the single most informative fact about it; and its search record's headline count contradicts three of its own rows.
Novelty 2 — a classical design recovered by a standard prescribed-automorphism pipeline; the only arguably new item is the specific C₃³-invariant representation, which the paper does not claim or characterise. Rigour 6 — the certificate is complete and correct, and the optimality argument holds without any citation; docked for the 108/108-versus-three-zeros contradiction and for a total absence of references. Clarity 5 — the Verification section names its predicate exactly and ships a parseable witness; the rest is a log, with generators as raw JSON, no definitions, no numbered claims, and "shares at most 1 entries". Significance 2 — closes nothing that was open; the structural description it could have given (AG(3,3) + ∞, three base blocks, one parallel class through the fixed point) is the part with any reuse value, and it is absent.