{"id":"65b1eabe-1f8f-40bb-bf90-8e813916ce2e","arxiv_id":"2607.07479","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal Schubert subspace codes of size A_q(u,ℓ,2(ℓ−t)) are constructed via direct-sum partial spreads with q-Johnson colorings and via field reduction of h-scattered subspaces.","lead":"The paper builds largest-possible constant-dimension codes that must meet a fixed subspace in high dimension while keeping pairwise intersections small. It gives two explicit constructions (direct-sum plus graph colorings, and field reduction from scattered spaces) that hit a simple counting upper bound in extremal regimes.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the only material limitation—the gap between the chromatic/clique necessary conditions of §3.2 and the partial-spread sufficient conditions of §3.1—and notes that the authors list it as open. That gap does not affect the validity of the theorems that are actually proved: whenever a sufficiently large partial spread exists, optimality is attained, and for h-scattered spaces the size is exactly the Gaussian binomial. Both constructions recover known special cases and the upper bound of Prop 2.5 is elementary. Consequently the ACCEPT / HIGH-confidence verdict stands; no adjustment is warranted.","tokens_in":21191,"tokens_out":453,"duration_ms":6665,"concrete_test":"Independently re-derive the intersection formula of Lemma 3.1 and the bijectivity argument of Thm 4.4 (using only the definition of h-scatteredness and field reduction) for the concrete parameters (q,n,k,u,ℓ,t)=(2,16,5,6,3,2); if both hold and |S_φ| equals the Gaussian binomial, the strongest claim is confirmed for a non-asymptotic instance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Thm 3.5 attaining the counting bound of Prop 2.5 via partial spreads + colorings of J_q(u,ℓ)^{k-t-1}, and Thm 4.4 attaining [u choose h]_q via field reduction of h-scattered spaces) rest on elementary, self-contained arguments. Lemma 3.1 correctly decomposes intersections under direct sum; the coloring argument and the bijection of Thm 4.4 via Lemma 4.3 are standard and free of hidden assumptions. The gap between sufficient (Thm 3.9 / Beutelspacher) and necessary (Cor 3.16 / Prop 3.19) conditions is real but is already stated as open problem (1) by the authors and does not undermine the theorems that are proved. No circularity, parameter fitting, or load-bearing error appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies Schubert subspace codes: constant-dimension codes whose codewords lie in a Schubert variety defined by a lower bound on intersection dimension with a fixed subspace U. For (ℓ,t)-intersecting sets with t ≤ ℓ−1 a counting argument yields the upper bound mq(n,k,u,ℓ,t) ≤ Aq(u,ℓ,2(ℓ−t)). The authors give two constructions that attain this bound in a range of parameters. The first uses a direct-sum decomposition Fnq = U ⊕ V, an optimal constant-dimension code in U, a partial (k−ℓ)-spread in V, and a coloring of the power Jq(u,ℓ)^{k−t−1}; necessary conditions via chromatic numbers and cliques are also derived. The second construction applies field reduction to evasive and h-scattered Fq-subspaces of Frqk, producing an (h,(h−1)k)-intersecting set of size exactly the Gaussian binomial [u choose h]q when the subspace is h-scattered, and recovering the earlier scattered-space construction of Alfarano–Rosenthal–Toesca as the case h=1.","tokens_in":21375,"tokens_out":918,"duration_ms":20025,"significance":"The work supplies the first systematic combinatorial constructions of optimal Schubert subspace codes beyond the single previously known family. The direct-sum approach cleanly reduces optimality to classical objects (partial spreads, chromatic numbers of powers of q-Johnson graphs) and makes the resulting chromatic and clique obstructions explicit; the field-reduction approach links the problem to the well-studied geometry of scattered and evasive subspaces and yields an exact size formula. All arguments are elementary linear algebra and graph theory, fully written out, and free of circularity or fitted parameters. The remaining gap between sufficient and necessary conditions is correctly identified as an open problem rather than papered over. The contribution is solid and of clear interest to the combinatorial coding-theory community.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 3.5 and the surrounding discussion it would help the reader to recall explicitly that the induced subgraph Γ = Jq(u,ℓ)^{k−t−1}[A] may have chromatic number strictly smaller than that of the full power graph; Remark 3.6 already notes this, but a short forward pointer in the theorem statement would make the strongest form of the result easier to apply.","section":null},{"comment":"Section 4, after Proposition 4.2: when h > 1 one has t = (h−1)k which typically exceeds ℓ−1 = h−1, so the counting bound of Proposition 2.5 does not apply. A single clarifying sentence at the beginning of the section would prevent a reader from expecting optimality claims that the authors correctly do not make (cf. Remark 4.7 and open problem (3)).","section":null},{"comment":"Corollary 3.8: the hypothesis u ≤ n/2 is used only to guarantee that the full set of 1-dimensional subspaces of V is large enough; it would be slightly cleaner to replace it by the weaker (and more transparent) inequality [u choose 1]q ≤ [n−u choose 1]q.","section":null},{"comment":"Example 3.21: the concrete numerical thresholds (n ≥ 16 even, n ≥ 17 odd, etc.) are useful; stating the precise Beutelspacher size used for each parity of n would make the verification fully self-contained.","section":null},{"comment":"Minor typographical points: “Universit` a” and “Universit´e” appear with inconsistent accent encoding; “K¨ otter” should be “Kötter” throughout; the arXiv identifier of reference [8] is listed as 2602.10777 (future-dated) and may need updating once the preprint is public.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is technically clean and the authors are appropriately cautious about the gap between necessary and sufficient conditions. I see no reason to request a major revision. The paper is a natural and well-executed sequel to the authors’ earlier work [1]; the self-citation pattern is justified by the subject matter."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper cleanly settles the maximum size of (ℓ,t)-intersecting Schubert subspace codes for t ≤ ℓ-1. The counting upper bound (Prop 2.5) is elementary: inject the intersections with the fixed U into a constant-dimension code inside U. Both constructions attain it for new parameter ranges.\n\nWhat is new: the direct-sum method that pairs an optimal code A in Gr_q(ℓ,u) with a partial (k-ℓ)-spread in a complement V, using a coloring of the power J_q(u,ℓ)^{k-t-1} so that same-color spaces can share a B (Thm 3.5). The necessary chromatic and clique conditions in §3.2 are original and correctly identify the obstructions. The field-reduction construction from h-scattered spaces (Thm 4.4) gives exact size [u choose h]_q and recovers the only prior example (ℓ=1,t=0) as the h=1 case. Lemma 3.1 (intersection splits under direct sum) and the bijection via Lemma 4.3 are clean linear algebra.\n\nSoft spots are real but limited and already flagged by the authors. The sufficient condition needs a partial spread at least as large as the chromatic number; existence is only asymptotic for large q (Thm 3.9) or via Beutelspacher, leaving a concrete gap with the necessary bounds (Cor 3.16, Prop 3.19). Example 3.21 shows the gap is not tiny. For h>1 the field-reduction codes meet the counting bound but the authors only prove a weaker upper bound (Prop 4.6), so optimality among all Schubert codes remains open. None of this breaks the theorems that are proved.\n\nMath is elementary and fully written out; citations are appropriate (self-cites to their earlier Schubert paper are necessary). No circularity, no free parameters. This is for people working on constant-dimension codes, q-analogs of Johnson graphs, or scattered/evasive subspaces. It deserves a serious referee and should be accepted after ordinary polishing. I would cite the constructions and the necessary conditions if I work in the area.","headline":"Solid combinatorial paper that settles the extremal size for Schubert subspace codes when t ≤ ℓ-1, with two clean constructions and an honest gap left open.","tokens_in":21977,"tokens_out":575,"would_cite":true,"duration_ms":5749,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B27","05C15","51E20"],"pacs":[],"model":"grok-4.5","headline":"Optimal Schubert subspace codes exist when a partial spread is large enough to color a power of the q-Johnson graph, and field reduction from scattered spaces hits the same bound exactly.","keywords":["Schubert subspace codes","constant-dimension codes","partial spreads","q-Johnson graphs","scattered subspaces","field reduction","evasive subspaces"],"falsifier":"For concrete parameters where the asymptotic inequality of Theorem 3.9 fails (for example u=6, ℓ=3, k=5 and n=12), exhibit an optimal (ℓ,t)-intersecting set of direct-sum type, or prove that every map from the full Grassmannian of ℓ-spaces into the Grassmannian of (k−ℓ)-spaces in V violates the intersection bound.","tokens_in":22091,"feed_emoji":"📐","tokens_out":844,"duration_ms":7319,"temperature":0.7,"pith_summary":"Schubert subspace codes are constant-dimension families of k-dimensional subspaces that must intersect a fixed u-dimensional subspace U in dimension at least ℓ, while any two distinct codewords intersect in dimension at most t. When t is at most ℓ−1 a simple counting argument caps their size by the ordinary constant-dimension code size A_q(u,ℓ,2(ℓ−t)). The paper supplies two explicit constructions that meet this upper bound. The first decomposes the ambient space as U ⊕ V, takes an optimal code of ℓ-spaces inside U, and completes each of them by a (k−ℓ)-space from V chosen according to a proper coloring of a power of the q-Johnson graph; the construction succeeds as soon as V contains a partial spread larger than that chromatic number. The second construction pulls back evasive or h-scattered spaces over an extension field via field reduction; for scattered spaces the resulting code has size exactly the Gaussian binomial coefficient [u choose h]_q. Together the two methods give the first systematic families of optimal Schubert codes beyond the single special case previously known.","feed_headline":"Optimal Schubert codes from spreads and colorings","feed_subtitle":"Partial spreads that color q-Johnson powers hit the counting bound; scattered spaces give the size exactly.","key_machinery":"Direct-sum maps φ that send each ℓ-space A ⊂ U to a (k−ℓ)-space φ(A) ⊂ V so that dim(A1 ∩ A2) + dim(φ(A1) ∩ φ(A2)) ≤ t; such maps exist precisely when the fibers of φ form independent sets of the power graph J_q(u,ℓ)^{k−t−1} and adjacent vertices receive pairwise disjoint images.","core_discovery":"When t ≤ ℓ−1 the maximal size m_q(n,k,u,ℓ,t) equals A_q(u,ℓ,2(ℓ−t)) whenever a complement V of U admits a partial (k−ℓ)-spread whose cardinality is at least the chromatic number of the (k−t−1)-st power of the q-Johnson graph J_q(u,ℓ). Independently, the field reduction of any h-scattered F_q-subspace of F_{q^k}^r produces an (h,(h−1)k)-intersecting set of size exactly [u choose h]_q inside the Grassmannian of hk-spaces.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Optimal Schubert codes via spreads and q-Johnson colorings","Schubert codes hit counting bounds with partial spreads","Exact sizes for Schubert codes from scattered subspaces","Max Schubert subspace codes via direct sums and colorings","Field reduction of scattered spaces yields Schubert codes"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The direct-sum optimality proofs need a partial spread in the complement that is large enough to color the relevant power of the q-Johnson graph; that size is guaranteed only asymptotically for large q under a dimensional inequality, or by classical spread bounds that leave a concrete gap with the necessary chromatic and clique conditions.","fun_headline_variants_meta":{"raw":{"variants":["Optimal Schubert codes via spreads and q-Johnson colorings","Schubert codes hit counting bounds with partial spreads","Exact sizes for Schubert codes from scattered subspaces","Max Schubert subspace codes via direct sums and colorings","Field reduction of scattered spaces yields Schubert codes"]},"model":"grok-4.5","effort":"low","cost_usd":0.007004,"raw_usage":{"total_tokens":1684,"prompt_tokens":727,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":70040000,"prompt_tokens_details":{"text_tokens":727,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":883,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":727,"tokens_out":74,"duration_ms":8891,"temperature":1.0,"reasoning_tokens":883,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T18:55:22.067623+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For concrete parameters where the asymptotic inequality of Theorem 3.9 fails (for example u=6, ℓ=3, k=5 and n=12), exhibit an optimal (ℓ,t)-intersecting set of direct-sum type, or prove that every map from the full Grassmannian of ℓ-spaces into the Grassmannian of (k−ℓ)-spaces in V violates the intersection bound.","supporting_citations":[],"review_version":2}