{"id":"39aee679-e064-4d8d-b5f4-691aa3db831e","arxiv_id":"2509.03186","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit families of F_q-linear QMDS codes over F_{q^h} achieve lengths q^h+2 and q^h+3, exceeding the classical linear MDS bound of q^h+1.","lead":"Researchers construct additive codes over finite fields with lengths exceeding the usual MDS bound, by using geometric packings of subspaces. The paper gives explicit families of QMDS codes of length q^h+2 and q^h+3, and studies when their duals remain optimal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the subspace-existence assumption in §4.2 is valid and constructible, so the central claim stands.","rationale":"The reader's conditional verdict stems from the missing proof of existence of Y_i and S_i in Section 4.2. My analysis shows these subspaces always exist for the stated parameter ranges. The Y_i can be built from coordinate hyperplanes because the dimension condition r1 ≤ 2h/3 ensures three (h−r1)-dimensional zero-sets cover all coordinates. The S_i exist because a partial r2-spread of size at least 3 is guaranteed by Beutelspacher's theorem whenever r2 ≤ h/2; the case r2 = h/2 still gives a spread of size q^{h/2}+1 ≥ 3. Thus Theorem 4.6 is valid as stated, modulo a short existence lemma that the authors should add. No fatal flaw or internal inconsistency was found in the central construction. The paper's main claim—long additive QMDS codes of length q^h+2 and q^h+3—is supported. Minor typographical issues in the dual-code section are not load-bearing. Therefore the reader's conditional verdict need not be changed; the paper is acceptable pending minor exposition improvements, but I find no significant objection that would alter the verdict.","tokens_in":14198,"tokens_out":40940,"duration_ms":397852,"concrete_test":"Verify the existence for a concrete allowed parameter set, e.g., h=6, q=2, r0=1, r1=4, r2=3. Explicitly construct Y1={x:x1=x2=0}, Y2={x:x3=x4=0}, Y3={x:x5=x6=0} in F_2^6 and check triple intersection is {0}. For S_i, take three blocks from a Desarguesian 3-spread in F_2^6 (which has 9 blocks) and verify pairwise trivial intersections. If both succeed, the assumption in Theorem 4.6 is satisfied.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only candidate load-bearing gap is the unproved existence in §4.2 of Y1,Y2,Y3 (r1-dimensional, triple intersection zero) and S1,S2,S3 (r2-dimensional, pairwise disjoint). This does not actually jeopardize the theorem. For Y_i, since r1 ≤ 2h/3, we have h−r1 ≥ h/3, so 3(h−r1) ≥ h; one can choose three subsets of {1,…,h} of size h−r1 whose union is the whole set, and define Y_i as the coordinate subspaces vanishing on those subsets. Their intersection is {0}. For S_i, r2 ≤ h/2 gives h = ar2+b with a ≥ 2, and Beutelspacher's bound (Theorem 2.3) provides at least q^{r2+b}+1 ≥ 3 pairwise disjoint r2-dimensional subspaces. Hence three such S_i always exist for every allowed q,h,r0,r2. The exclusion h=7 is exactly where no integer r1,r2 satisfy the dimension sum, confirming the authors' awareness of the constraints. The proof of Theorem 4.6 is otherwise a routine Vandermonde argument, and the resulting packing gives a genuine QMDS code of length q^h+3. Section 4.1's q^h+2 construction is also sound. Minor textual issues in Section 5 do not affect the main construction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs additive F_q-linear codes in F_qh^n attaining the generalized Singleton bound (QMDS) with length exceeding q^h+1. Section 3 recalls the geometric translation between additive codes and subspace packings. Section 4.1 gives a packing of q^h+2 subspaces of dimension (k-2)h+r0 in F_q^{(k-1)h+r0}, with a Vandermonde proof that any k meet trivially; Section 4.2 extends this to q^h+3 under hypotheses h≥6, h≠7 and r1+r2=h+r0; Section 4.2.1 gives a k=3 variant with q^h+g(q) blocks; Section 4.3 treats k=2 via partial spreads. Section 5 studies dual codes and shows some constructed codes are not dually QMDS; Section 6 connects dually QMDS codes to dimensional dual arcs and derives nonexistence for odd q in extremal cases.","tokens_in":14563,"tokens_out":30194,"duration_ms":290517,"significance":"If the gaps below are patched, the paper provides explicit long additive QMDS codes beyond the q^h+1 linear MDS bound, with elementary and verifiable arguments. The main construction is a genuine geometric contribution: it reinterprets the QMDS condition as a (k-1)-packing and uses Vandermonde systems to ensure vanishing intersections. The paper also spells out the relation between dual hyperovals and dually QMDS codes. The proofs are explicit and do not rely on fitted parameters or circular reasoning; external results are used as tools. The dual-code section is useful but contains the correctness issues noted below.","major_comments":[{"comment":"The proof of Theorem 4.6 is conditional on the asserted existence of Y1,Y2,Y3 (r1-dimensional, triple intersection zero) and S1,S2,S3 (r2-dimensional, pairwise disjoint) for every allowed q,h,r0,r1,r2. This is load-bearing for the length q^h+3 construction. Existence is true and can be shown by coordinate subspaces for the Y_i (three subsets of [h] of size h-r1 with union [h]) and by Beutelspacher's bound for the S_i, but the manuscript should state this explicitly; as written it is an unstated assumption.","section":"§4.2, before Theorem 4.6"},{"comment":"The step 'Since r2 ≥ h−r1, we have #Ω2 ≤ Γ' is not justified as written. If Γ is only an inclusion-maximal partial (h−r1)-spread, the inequality is false in general; a maximum spread is needed. The intended argument is to choose an (h−r1)-subspace inside each S_i; these are pairwise disjoint, so #Γ ≥ #Ω2, and then Γ^⊥ gives #Ω1 ≥ #Γ. Please include this argument and use #Γ rather than Γ.","section":"§4.2.1"},{"comment":"The statement needs k≥3: for k=2, A_{q,h,2} is a partial-spread code and Corollary 5.9 implies its dual is QMDS, so the theorem as stated is false. Also, in the last paragraph 'let C = B^⊥_{q,h}' must be 'let C = B_{q,h}' for Theorem 5.6 to apply. Finally, the inequality dim(W∞1∩W∞2) ≥ (k−3)h+2r0 in the B_{q,h,k} case follows from dim(Y1∩Y2) ≥ 2r1−h ≥ 2r0, but this is not said; the reader is left with an unexplained jump.","section":"Theorem 5.7"}],"minor_comments":[{"comment":"The theorem assumes t∤r and then says the bound is tight when t divides r; state the t|r case separately to avoid confusion.","section":"Theorem 2.2"},{"comment":"Replace 'maximal' by 'maximum' for Γ and Ω1, and define g(q)=#Ω2 after Ω2 is introduced.","section":"§4.2.1"},{"comment":"The symbol n is used both for the length and in n−1−r0/h; use |Ω| or specify n = q^h+f(q) explicitly.","section":"Corollary 5.9"},{"comment":"State the lower bound 1 ≤ r0 in the parameter conditions; the text only says r0 ≤ h/2.","section":"Section 4.1"},{"comment":"The notation A^⊥_{q,h,k}, B^⊥_{q,h,k}, B^⊥_{q,h} should be accompanied by the intended parameter ranges (especially k≥3) so the statement is unambiguous.","section":"Theorem 5.7"}],"recommendation":"major_revision","confidential_remarks":"The main construction appears sound and the central existence claim for long QMDS codes is very likely correct. The paper needs a short existence lemma for the subspaces in §4.2 and a rigorous inequality in §4.2.1, plus corrections to Theorem 5.7. The k=2 'long' codes in §4.3 overlap with known partial-spread constructions, so the novelty rests mainly on Sections 4.1 and 4.2; the authors should make that novelty explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the main constructions are sound, and the one apparent gap in Section 4.2 — existence of the auxiliary subspaces — is not actually a gap. You can build Y_1,Y_2,Y_3 as coordinate subspaces vanishing on complementary subsets of {1,...,h}, and pull S_1,S_2,S_3 from a Beutelspacher partial spread; the inequalities r1≤2h/3 and r2≤h/2 make it work. So both the q^h+2 and q^h+3 families stand.\n\nWhat is genuinely new: the explicit Vandermonde packings give additive QMDS codes of length q^h+2 and q^h+3, beyond the usual q^h+1 ceiling for F_{q^h}-linear MDS codes. The packing viewpoint is clean, and the dual-code analysis in Section 5 gives usable conditions for the QMDS property to survive duality. The paper is also honest about provenance: Section 4.3 reproduces [1, Thm 23] in the r0|h case, and Theorem 5.10 is a new proof of [14, Thm 1]. That overlap is not concealed.\n\nSoft spots, in proportion. First, Theorem 4.6 leans on 'same as Theorem 4.3' for cases with two W∞'s. The math works, but the proof is terser than it should be; the key point is that even when Y_1∩Y_2 is nontrivial, the Vandermonde system kills the y coordinate. Second, Theorem 5.7 has a real typo: the final case needs C = B_{q,h}, not C = B^⊥_{q,h}. The dimension computation only makes sense for B. That is minor but should be fixed. Third, the chain #Ω1 ≥ #Γ ≥ #Ω2 in Section 4.2.1 is compressed; it is recoverable by injecting (h−r1)-subspaces into the r2-spread and dualizing, but a skeptical reader will want a sentence.\n\nI disagree with the reader's strongest concern. The §4.2 existence assumption is not load-bearing; the stress-test construction is valid, and the authors' exclusion of h=7 matches exactly where the dimension constraints cannot be satisfied. So the conditional verdict should resolve to accept the main claims.\n\nThis paper is for coding theorists and finite geometers working on additive codes or subspace packings. It deserves a serious referee. I would send it to review, asking for the Theorem 5.7 typo fix and a small expansion of the terse cases.","headline":"The q^h+2 and q^h+3 additive QMDS constructions are sound; the apparent §4.2 existence gap is fillable, so the paper deserves a normal referee.","tokens_in":15017,"tokens_out":10407,"would_cite":true,"duration_ms":101563,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","51E20","05B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Additive codes can reach lengths q^h+2 and q^h+3—beyond the classical q^h+1 barrier—while still meeting the generalized Singleton bound.","keywords":["additive codes","quasi-MDS codes","subspace packings","partial spreads","dual hyperovals","Vandermonde determinant","Singleton bound","long codes"],"falsifier":"Pick an allowed small parameter set, for instance q=2, h=6, r0=1, r1=4, r2=3, k=3. Explicitly search F_2^6 for three 4-dimensional subspaces Y1,Y2,Y3 with Y1∩Y2∩Y3={0} and three pairwise disjoint 3-dimensional subspaces S1,S2,S3; if they exist, build the code of Theorem 4.6 and compute its minimum distance by exhaustive enumeration of all 2^13 codewords. Any k-fold intersection in L other than {0}, or a minimum distance smaller than n-k+1 = 65, refutes the construction; a failed subspace search shows the q^h+3 claim has no content for that parameter set.","tokens_in":14151,"feed_emoji":"📐","tokens_out":17645,"duration_ms":166445,"temperature":0.7,"pith_summary":"The paper asks whether codes over F_{q^h} that are linear only over the smaller field F_q—'additive' codes, whose normalized dimension r/h may be fractional—can be longer than the longest classical F_{q^h}-linear MDS codes (length q^h+1) without losing the optimal Singleton distance. It answers yes, constructing explicit 'long' additive QMDS codes of length q^h+2 for all q≥2, h≥2, and length q^h+3 whenever certain auxiliary subspaces of prescribed dimensions exist. The construction works geometrically: a set of hyperplane-like subspaces in an F_q-vector space is shown to be a (k-1)-packing—any k of them meet only at zero—which by a known correspondence yields exactly the desired code parameters. A reader should care because the result shows the classical length bound is an artifact of full F_{q^h}-linearity, and the fractional slack can be spent on extra coordinates; it also connects the extremal dually QMDS codes to dual hyperovals, a central object in finite geometry.","feed_headline":"Additive codes beat the Reed–Solomon length limit","feed_subtitle":"Subspace-packing construction yields Singleton-optimal codes of length q^h+2 and q^h+3.","key_machinery":"The load-bearing identity is the Vandermonde determinant: for distinct α_1,...,α_k ∈ F_q^h, the system ∑ α_j^{i-1} x_i = 0 has only the trivial solution because det VM(α_1,...,α_k) = ∏_{i<j}(α_j - α_i) ≠ 0. The construction shapes every k-fold intersection so that it reduces to such a system—in the cases involving W_{∞1} or W_{∞2}, the special form of those subspaces first kills one or two of the field coordinates, shrinking the Vandermonde system—and the final r0 coordinates are killed by the F_q-linear independence of the basis elements 1, ξ, ..., ξ^{r0-1}. Thus the whole (k-1)-packing property is a statement about evaluating polynomials at distinct points.","core_discovery":"On its own terms, the central claim: the set L = {W_α} ∪ {W_{∞1}, W_{∞2}}, with W_α defined by a Vandermonde-type equation ∑ α^{i-1} x_i + α^{k-1} x_k = 0, is a (k-1)-packing—any k of its subspaces intersect only at zero. The proof reduces each case to a Vandermonde system with distinct nodes, whose determinant is nonzero, forcing the field coordinates to zero; the final r0 coordinates then vanish because 1, ξ, ..., ξ^{r0-1} are F_q-linearly independent. This packing corresponds to an additive [q^h+2, (k-1)+r0/h, q^h+3-k]^h_q QMDS code, which is longer than q^h+1. With three W_∞ subspaces and auxiliary families satisfying r1+r2 = h+r0, the length becomes q^h+3; the paper also proves the dual","pith_inferences":["The paper's upper bound n ≤ k-2+q^h+(q^h-1)/(q^{r0}-1) suggests that taking r0=1 could permit lengths on the order of q^h + q^{h-1}; whether the packing constructions can be pushed to reach that regime for general k is an open question, but the k=2 and k=3 families already show that the 'long' phenomenon is not an isolated accident.","The q^h+3 construction hinges on an existence assumption for auxiliary subspaces; investigating exactly which (h, r0, r1, r2) admit such triples—and why h=7 is the only excluded small case—looks like a tractable finite-geometry problem whose answer would either complete or bound the family.","The equivalence with dual hyperovals means that classifying h-dimensional dual hyperovals would immediately settle the longest possible dually QMDS codes with fractional dimension (k-1)+1/h; since dual hyperovals are known only in even characteristic, the odd-q nonexistence is probably a shadow of a deeper parity constraint on such extremal codes.","Because the k=2 partial-spread codes are dually QMDS, they may be the more practical family for applications such as quantum stabilizer codes, where both the code and its dual need good distance; the non-dually-QMDS long codes of Section 4.1 might still be useful when only one direction matters."],"forward_implications":["For every prime power q, every h ≥ 2, and every k with 2 ≤ k ≤ q^h - 1, choosing any r0 with 1 ≤ r0 ≤ h/2 gives an explicit additive QMDS code of length q^h+2 and dimension (k-1)h+r0 over F_q.","When h ≥ 6, h ≠ 7 and r0 ≤ h/6, the same idea yields length q^h+3, with r1 and r2 chosen so that r1+r2 = h+r0, provided the required auxiliary subspaces exist.","For k=3 the number of 'infinite' subspaces can be increased to the size of a large partial spread, giving length q^h + g(q) with g(q) at least ∑_{i=1}^{a-1} q^{i r2+b} + 1; for k=2, length q^h + f(q) with f(q) at least ∑_{i=1}^{a-1} q^{i r0+b} + 1, and exactly (q^h-1)/(q^{r0}-1) when r0 divides h.","The duals of the long codes from Sections 4.1 and 4.2 are not QMDS, so the long codes are examples of fractional QMDS codes that are not dually QMDS.","If a faithful additive QMDS code has r = h + r0, then its dual is also QMDS; in particular, the partial-spread construction with k=2 gives dually QMDS codes of length q^h+f(q) and distance 2."],"supporting_citations":[{"why":"Establishes the correspondence between additive codes and h-(n,r,d)_q systems/subspace packings, the generalized Singleton bound framework, and the upper bound n ≤ k-2+q^h+(q^h-1)/(q^{r0}-1) that defines 'long' codes.","marker":"[1]"},{"why":"Supplies the lower bound on partial spreads (Theorem 2.3) used to estimate the achievable growth f(q) and g(q) in the k=2 and k=3 constructions.","marker":"[3]"},{"why":"Proves the generalized Singleton bound d ≤ n - ceil(r/h) + 1 for F_q-linear F_{q^h}-codes, the inequality the QMDS definition attains.","marker":"[10]"},{"why":"Introduces the folded-Hamming-metric viewpoint and the quasi-MDS property; Theorem 5.10 of the paper is a re-proof of its Theorem 1, and the dual-hyperoval connection builds on its framework.","marker":"[14]"},{"why":"Proves that d-dimensional dual hyperovals exist only in even characteristic (Theorem 2.8), which yields the nonexistence results for extremal dually QMDS codes when q is odd.","marker":"[6]"},{"why":"Provides an explicit h-dimensional dual hyperoval in F_2^{2h+1} (Example 2.5), which through Corollary 6.1 gives a dually QMDS code matching the known parameter set.","marker":"[5]"},{"why":"Shows that additive QMDS codes that are not dually QMDS must be fractional, framing the dual-code results in Section 5.","marker":"[16]"}],"fun_headline_variants":["Additive codes break the q^h+1 length barrier","New QMDS codes exceed q^h+1 via subspace packing","Additive structure enables longer Singleton-optimal codes","Length q^h+2 and q^h+3 QMDS codes from packing","Beyond Reed-Solomon: additive codes push length limit"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The length q^h+3 construction assumes, for every parameter choice allowed by the theorem, that three F_q-subspaces of the required dimension r1 exist with triple intersection zero, and three pairwise disjoint F_q-subspaces of dimension r2 exist as well. The paper does not prove this existence; for h=7 the dimensions cannot even be chosen (which is why h=7 is excluded), so the q^h+3 construction stands or falls with this unproved existence.","fun_headline_variants_meta":{"raw":{"variants":["Additive codes break the q^h+1 length barrier","New QMDS codes exceed q^h+1 via subspace packing","Additive structure enables longer Singleton-optimal codes","Length q^h+2 and q^h+3 QMDS codes from packing","Beyond Reed-Solomon: additive codes push length limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000268,"raw_usage":{"total_tokens":1499,"prompt_tokens":830,"completion_tokens":669,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":579}},"tokens_in":574,"tokens_out":669,"duration_ms":7032,"temperature":1.0,"reasoning_tokens":579,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:09:41.089584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick an allowed small parameter set, for instance q=2, h=6, r0=1, r1=4, r2=3, k=3. Explicitly search F_2^6 for three 4-dimensional subspaces Y1,Y2,Y3 with Y1∩Y2∩Y3={0} and three pairwise disjoint 3-dimensional subspaces S1,S2,S3; if they exist, build the code of Theorem 4.6 and compute its minimum distance by exhaustive enumeration of all 2^13 codewords. Any k-fold intersection in L other than {0}, or a minimum distance smaller than n-k+1 = 65, refutes the construction; a failed subspace search shows the q^h+3 claim has no content for that parameter set.","supporting_citations":[{"cited_title":"and Popatia, T","cited_arxiv_id":null,"evidence_quote":"Establishes the correspondence between additive codes and h-(n,r,d)_q systems/subspace packings, the generalized Singleton bound framework, and the upper bound n ≤ k-2+q^h+(q^h-1)/(q^{r0}-1) that defines 'long' codes."},{"cited_title":"Partial spreads in finite projective spaces and partial designs","cited_arxiv_id":null,"evidence_quote":"Supplies the lower bound on partial spreads (Theorem 2.3) used to estimate the achievable growth f(q) and g(q) in the k=2 and k=3 constructions."},{"cited_title":"Linear codes in the folded Hamming distance and the quasi MDS property","cited_arxiv_id":"2406.13355","evidence_quote":"Introduces the folded-Hamming-metric viewpoint and the quasi-MDS property; Theorem 5.10 of the paper is a re-proof of its Theorem 1, and the dual-hyperoval connection builds on its framework."},{"cited_title":"On d-Dimensional Dual Hyperovals","cited_arxiv_id":null,"evidence_quote":"Proves that d-dimensional dual hyperovals exist only in even characteristic (Theorem 2.8), which yields the nonexistence results for extremal dually QMDS codes when q is odd."},{"cited_title":"On Generalized k-Arcs in PG(2n, q)","cited_arxiv_id":null,"evidence_quote":"Provides an explicit h-dimensional dual hyperoval in F_2^{2h+1} (Example 2.5), which through Corollary 6.1 gives a dually QMDS code matching the known parameter set."},{"cited_title":"Some new classes of additive MDS and almost MDS codes over finite fields","cited_arxiv_id":null,"evidence_quote":"Shows that additive QMDS codes that are not dually QMDS must be fractional, framing the dual-code results in Section 5."}],"review_version":1}