{"id":"c2bd4bfc-4364-404a-bbcb-6bb37f883e2e","arxiv_id":"2607.26684","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A reduction from QMDS subspace families to partial spreads yields tighter length upper bounds than the Ball–Lavrauw–Popatia Griesmer-type bound in several regimes.","lead":"The paper proves that length bounds for quasi-MDS codes reduce to classical partial-spread size bounds in finite geometry. This recovers a known Griesmer-type bound and tightens it in several parameter ranges using Drake–Freeman, Năstase–Sissokho, and vector-space-partition estimates.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the only soft spot: Theorem 6 supplies a necessary local condition, so the derived length bounds need not be tight (Remark 25). That does not threaten validity of the upper bounds or the claim that they improve Ball et al. in several regimes. Proofs are elementary and the external geometric theorems are quoted with matching hypotheses. No further load-bearing gap appears. Verdict remains ACCEPT; no adjustment required.","tokens_in":20128,"tokens_out":561,"duration_ms":36400,"concrete_test":"Pick concrete parameters where Năstase–Sissokho applies with m=e-1, e.g. q=2, r0=4, ρ=2, r=6, e=2 (so k=2·6+4=16). Compute the numerical value of the Theorem 8 bound and of the integer packing bound of Corollary 28; confirm the claimed improvement ΔZ_NS=q^ρ-1=3. Separately, enumerate (or cite a known construction of) a partial 4-spread in F_2^{10} of size equal to the Năstase–Sissokho value and verify it saturates μ_2(10,4), confirming the imported constant is sharp.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 6 (reduction of type-[n,s,k,e] families with k=er+r0 to partial r0-spreads) plus the imported Drake–Freeman / Năstase–Sissokho / Honold–Kiermaier–Kurz bounds yielding Theorems 7–9. The proof of Theorem 6 is standard linear algebra: after fixing m members, the (e-m)-fold residual intersections inside a (k-mr)-dimensional slice form a genuine partial r0-spread (pairwise trivial intersection follows because a nonzero common vector would lie in ≥e+1 original members). Distinct index sets produce distinct subspaces because dim r0>0 and intersections are trivial, so the binomial count is exact. Corollary 11 correctly lifts arbitrary QMDS codes to faithful families, so length upper bounds transfer. Remark 25 correctly flags that the local condition is necessary-not-sufficient and that the resulting bounds may be loose; the authors do not claim tightness. No internal inconsistency, hidden assumption, or misapplication of the external partial-spread theorems was found. The improvements over the Ball–Lavrauw–Popatia packing bound are therefore valid (if sometimes strict) upper bounds in the stated regimes.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies upper bounds on the length n of F_q-linear quasi-MDS (QMDS) codes of type [n,r,k,d] in the folded Hamming metric. It associates generator matrices with families of subspaces of F_q^k (kernels of the column blocks) and shows that the QMDS condition translates into an intersection condition, yielding faithful families of type [n,k-r,k,⌈k/r⌉-1]. The central technical contribution is Theorem 6: for k=er+r_0 with 0<r_0<r, every m∈{0,…,e-1} satisfies binom(n-m,e-m)≤μ_q(k-mr,r_0), where μ_q is the maximum size of a partial r_0-spread. Importing the packing bound recovers the Griesmer-type bound of Ball–Lavrauw–Popatia; importing Drake–Freeman, Năstase–Sissokho and Honold–Kiermaier–Kurz then produces the stricter length bounds of Theorems 7–9 in several regimes. The correspondence (Proposition 10, Corollary 11, Theorems 1–2) and the reduction are developed carefully, with an explicit warning (Remark 25) that the local condition is only necessary.","tokens_in":20419,"tokens_out":813,"duration_ms":16336,"significance":"The work cleanly links the length problem for fractional MDS / QMDS codes to the well-developed theory of partial spreads and 1-subspace packings. The reduction of Theorem 6 is elementary linear algebra yet immediately yields concrete numerical improvements over the best previously published general bound (Ball et al.), as quantified by the integer deltas Δ_DF, Δ_NS and Δ_VSP and the accompanying tables. Because the imported geometric theorems are classical and correctly specialized, the new upper bounds are reliable and ready for use by coding theorists. The paper is therefore a solid, incremental contribution that strengthens the geometric toolkit available for folded-Hamming and additive codes.","major_comments":[],"minor_comments":[{"comment":"The exclusion of the divisible case s|k (and r|k) is stated repeatedly but never collected in one place; a single sentence in §2.2 or after Definition 19 would improve readability.","section":"§2.2 / §3"},{"comment":"In the proof of Theorem 6 the ambient space U of dimension k-mr is chosen inside the intersection of the m fixed members; it would help the reader to note explicitly that the choice of U does not affect the subsequent partial-spread bound.","section":"Theorem 6"},{"comment":"Tables comparing Δ_DF, Δ_NS and Δ_VSP (pp. 16–21) are useful but lack captions and are not numbered; numbering them and adding a one-line caption would make cross-reference easier.","section":"§5"},{"comment":"A few typographical inconsistencies appear (e.g., “QMDScodes”, missing spaces around em-dashes, “Năstase” vs “N\\u{a}stase”). A light copy-edit pass would remove them.","section":null},{"comment":"Reference [4] (Bartoli et al., arXiv:2509.03186) is cited for a related correspondence; a brief clarifying sentence on how the present generator-side families differ from that work would be helpful.","section":"§2.3"}],"recommendation":"accept","confidential_remarks":"The manuscript is technically sound and the improvements, while sometimes modest, are genuine. Fit for a solid coding-theory or finite-geometry journal is clear; I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is Theorem 6: after fixing m members of a type-[n,s,k,e] family with k=er+r0, the residual (e-m)-fold intersections inside a (k-mr)-slice form a genuine partial r0-spread, so the binomial count is at most μq(k-mr,r0). That single inequality lets them pull in Drake–Freeman, Năstase–Sissokho and Honold–Kiermaier–Kurz and produce Theorems 7–9, which improve the Ball–Lavrauw–Popatia packing bound by explicit integer corrections (⌊ω⌋+1, qρ-1, or the min of those with the VSP tail term).\n\nWhat is new is precisely that reduction and the resulting tables of improvements. The correspondence between folded-Hamming generator matrices and 1-subspace packings (Prop. 10, Cor. 11, Thms 1–2) is clean linear algebra and matches the earlier packing literature; recovering Ball et al. as the m=e-1 packing case is expected and correctly done. Proofs are explicit dimension counts; external theorems are quoted with the right hypotheses. Self-citations are to their own prior QMDS paper and are not load-bearing for the new bounds.\n\nThe soft spot is exactly the one they name in Remark 25: the reduction only enforces a local necessary condition, so the bounds can be strictly loose (their F2 example attains the local spread bound yet fails the global type). That is not hidden and does not invalidate the upper bounds; it just means one should not expect equality cases in general. Divisible cases are excluded for the usual spread-existence reasons; no other gaps showed up on a second pass.\n\nThis is for people who already care about length tables for fractional/QMDS codes or who keep partial-spread bounds on the shelf. Solid within-subfield contribution, formally grounded, no circularity. I would send it to referees without hesitation.","headline":"Clean geometric reduction that legitimately tightens QMDS length bounds by importing classical partial-spread results; the necessary-not-sufficient caveat is already flagged by the authors and does not break the claims.","tokens_in":21060,"tokens_out":511,"would_cite":true,"duration_ms":11089,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","51E23","05B25"],"pacs":[],"model":"grok-4.5","headline":"Quasi-MDS codes cannot be longer than a partial-spread bound imported from finite geometry.","keywords":["QMDS codes","folded Hamming distance","subspace packings","partial spreads","vector space partitions","fractional MDS codes","Griesmer-type bounds"],"falsifier":"Exhibit a QMDS code whose length exceeds the integer bound of Theorem 7, 8 or 9 for some admissible q, r, r_0, e, or prove that no such code exists by constructing a matching family of subspaces that saturates the partial-spread number.","tokens_in":21002,"feed_emoji":"📐","tokens_out":1024,"duration_ms":22112,"temperature":0.7,"pith_summary":"The paper asks how long an F_q-linear quasi-MDS (QMDS) code can be in the folded Hamming metric before its length is forced by the field size q. It translates a generator matrix into a family of subspaces whose intersection pattern encodes the minimum distance, then proves that any such family is controlled by the size of certain partial spreads. Importing the sharpest known partial-spread bounds immediately recovers the earlier Griesmer-type length bound and, in several concrete regimes, improves it by an explicit integer correction that grows with the remainder parameters. A sympathetic reader cares because QMDS codes are the nearest relatives of classical MDS codes once the dimension is not a multiple of the fold size; tighter length ceilings therefore limit how far fractional Singleton-attaining codes can exceed the classical MDS conjecture.","feed_headline":"QMDS length capped by partial-spread geometry","feed_subtitle":"A reduction to finite-geometry packings tightens the Ball–Lavrauw–Popatia ceiling in several regimes","key_machinery":"Theorem 6, the reduction from 1-subspace packings to partial spreads: after fixing any m members of the family, the remaining (e-m)-fold intersections form a partial r_0-spread inside a subspace of dimension k-m r, forcing the binomial count of those intersections to be at most μ_q(k-m r, r_0).","core_discovery":"For a faithful family of subspaces of type [n, s, k, e] arising from a QMDS code with k = e r + r_0 (0 < r_0 < r), every integer m between 0 and e-1 satisfies the binomial inequality binom(n-m, e-m) ≤ μ_q(k-m r, r_0), where μ_q(v, t) is the maximum size of a partial t-spread in F_q^v. Substituting the Drake–Freeman, Năstase–Sissokho or vector-space-partition bounds for μ_q yields length upper bounds that recover the Ball–Lavrauw–Popatia Griesmer-type bound and strictly improve its integer form in several parameter ranges.","pith_inferences":["Because the reduction is only necessary, the true maximal length may be strictly smaller than every bound obtained from Theorem 6; the gap is already visible in the binary example of Remark 25.","The same reduction applies verbatim to any 1-(k,s,e)_q packing, so the length tables immediately supply new upper bounds for subspace-packing numbers A_q(k,s,1;e).","When r_0 divides r the exceptional packing case collapses back to the classical Griesmer expression, suggesting that the genuinely new improvements live only when the remainder is nonzero."],"forward_implications":["The Ball–Lavrauw–Popatia Griesmer-type length bound is recovered simply by feeding the ordinary packing bound into the reduction with m = e-1.","Whenever the Năstase–Sissokho hypothesis holds, the length ceiling improves the packing-derived integer bound by exactly q^ρ-1.","Outside that regime the vector-space-partition correction min(q^ρ-1,(q-1)(r_0-1)) still improves the packing bound and, for intermediate ρ, can beat Drake–Freeman.","All improved ceilings remain asymptotic to e-1 + q^r plus a lower-order term, so the dominant growth in q is unchanged."],"fun_headline_variants":["QMDS lengths bounded by partial-spread sizes via subspace reduction","Subspace families tie QMDS length caps to partial t-spread maxima","Drake–Freeman bounds tighten QMDS length limits in key regimes","Finite-geometry packings sharpen Ball et al. QMDS ceilings","Partial spreads recover and improve Griesmer-type QMDS bounds"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The reduction only forces a local intersection condition after m members are fixed; a family can meet every local partial-spread bound and still fail the global intersection requirement that defines the code.","fun_headline_variants_meta":{"raw":{"variants":["QMDS lengths bounded by partial-spread sizes via subspace reduction","Subspace families tie QMDS length caps to partial t-spread maxima","Drake–Freeman bounds tighten QMDS length limits in key regimes","Finite-geometry packings sharpen Ball et al. QMDS ceilings","Partial spreads recover and improve Griesmer-type QMDS bounds"]},"model":"grok-4.5","effort":"low","cost_usd":0.004672,"raw_usage":{"total_tokens":1335,"prompt_tokens":780,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":46724000,"prompt_tokens_details":{"text_tokens":780,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":462,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":780,"tokens_out":93,"duration_ms":11154,"temperature":1.0,"reasoning_tokens":462,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T00:03:46.236103+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a QMDS code whose length exceeds the integer bound of Theorem 7, 8 or 9 for some admissible q, r, r_0, e, or prove that no such code exists by constructing a matching family of subspaces that saturates the partial-spread number.","supporting_citations":[],"review_version":1}