{"id":"e429668e-0284-4ede-a710-619c7310c318","arxiv_id":"1908.03239","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sum-rank Hamming codes are the longest one-error-correcting codes in the sum-rank metric; when the extension degree is one they are characterized by maximal partial spreads and can be perfect.","lead":"This paper creates error-correcting codes for messages split into blocks, where an error is measured by matrix rank, and shows these codes are equivalent to a known geometric packing problem in a simple case. The same codes work for networked communication and for storage systems over tiny fields like binary.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's claimed bijection fails for q=2, r=3, N=2: a maximal-size partial 2-spread exists but no proper sum-rank Hamming code does.","rationale":"The reader's weakest assumption identified the omitted proof of the unequal-sublength (improper) isometry case as the main risk. That is a legitimate concern, but it is secondary to a more serious internal flaw: even in the proper equal-sublength case, the claimed bijection with maximal-size partial spreads is false for some parameters. The counterexample q=2, r=3, N=2 is simple and decisive: a maximal-size partial 2-spread exists in F_2^3, but no proper sum-rank Hamming code of redundancy 3 exists because at least three pairwise disjoint 2D subspaces would be needed and any two 2D subspaces of F_2^3 intersect. The error is that maximality by cardinality does not guarantee that the partial spread spans the ambient space, while a redundancy-r code requires a full-rank parity-check matrix. This affects the central claim advertised in the abstract and in Theorem 4, not just a corner case: for any N with r/2 < N < r, the maximal partial N-spread has size 1 and fails to span, while no proper Hamming code exists. The perfect-code corollary (Corollary 15) is restricted to N dividing r and may remain correct, and the construction from spreads that do span is valid, but the unrestricted classification is not. Since the main theorem of the paper is false as stated, the appropriate verdict is REJECT rather than CONDITIONAL; a corrected version would need to add a spanning condition and restrict the bijection accordingly.","tokens_in":56,"tokens_out":21559,"duration_ms":484385,"concrete_test":"Analytic check of the case q=2, r=3, N=2: (1) Verify from Proposition 9 or directly that no linear code with parity-check matrix H in F_2^{3 x n}, n=2ℓ, with each block H_i of size 3x2 having rank 2 and pairwise disjoint column spaces, exists for any ℓ≥2, because any two 2D subspaces of F_2^3 intersect nontrivially. (2) Note that ℓ=1 gives d_SR≤1<3. (3) Observe that a single 2D subspace is a maximal-size partial 2-spread in F_2^3. This falsifies the bijection in Theorem 4, since the spread set is nonempty while the code class is empty. If the authors intend a spanning condition, the test is to check whether the claimed bijection in Theorem 4 holds when the partial spread is required to span F_q^r; the counterexample above should be excluded by that revised hypothesis.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central proper-case claim, Theorem 4, asserts a bijection between isometry classes of proper sum-rank Hamming codes of redundancy r and maximal-size partial N-spreads in F_q^r. This fails when a maximal-size partial N-spread does not span F_q^r, because a parity-check matrix H of a code with redundancy r must have rank r, so its column spaces must collectively span F_q^r. Maximality by cardinality does not imply spanning. Concrete counterexample: q=2, r=3, N=2. Any two distinct 2-dimensional subspaces of F_2^3 intersect in at least one nonzero vector, so a partial 2-spread has maximum size 1; a single plane is a maximal-size partial 2-spread. But no proper sum-rank Hamming code with these parameters exists. When m=1, any nonzero codeword has weight at most the number of blocks ℓ, so d_SR(C)≥3 forces ℓ≥3. A parity-check matrix with ℓ≥3 blocks would need ℓ pairwise disjoint 2D column spaces in F_2^3, which is impossible for ℓ≥2 by the intersection bound. Thus one side of the claimed bijection is empty while the other is nonempty. The root cause is the missing requirement that the partial spread span F_q^r, or equivalently that the constructed H have full row rank r. This is a substantive gap in the proof of Theorem 3 and the bijection in Theorem 4, not merely a deferred case. The reader's concern about the omitted unequal-sublength extension of Theorem 2 is real, but the proper case already contains a false statement for parameters with N>r/2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces sum-rank Hamming codes, defined as the longest linear codes with minimum sum-rank distance at least 3 for a fixed redundancy, base field, and extension degree, together with their duals, sum-rank simplex codes. General bounds on code length and number of shots are given. For extension degree m=1, the paper characterizes proper sum-rank Hamming codes via maximal-size partial spreads, proves they are perfect when the sublength divides the redundancy, provides a syndrome decoding algorithm, and derives bounds for simplex codes. Applications to multishot matrix-multiplicative channels and binary locally repairable codes are also presented.","tokens_in":18074,"tokens_out":7596,"duration_ms":80277,"significance":"If the m=1 classification were correct as stated, it would be an elegant and useful bridge between single-error sum-rank correction and finite-geometry partial spreads, with practical benefits including an efficient decoding algorithm and explicit binary LRC constructions. The perfect-code result for N dividing r is a natural sum-rank analogue of the classical Hamming-code perfectness. However, the central bijection is false without an additional spanning condition, and several load-bearing proofs for the unequal-sublength (improper) case are omitted. The core idea is salvageable, but the manuscript needs substantive revision before the claims are supported.","major_comments":[{"comment":"The stated equivalence between proper sum-rank Hamming codes and maximal-size partial N-spreads is false when a maximal-size partial N-spread does not span F_q^r. For q=2, r=3, N=2, a maximal-size partial 2-spread in F_2^3 has size 1 (any two distinct 2-dimensional subspaces intersect nontrivially), so it exists. But no proper sum-rank Hamming code with these parameters exists: since m=1, any nonzero codeword has sum-rank weight at most the number of blocks, so d_SR(C)>=3 forces at least 3 blocks; meanwhile a parity-check matrix with redundancy r=3 would need three pairwise disjoint 2-dimensional column spaces in F_2^3, which is impossible. Equivalently, a parity-check matrix assembled from a single 2-dimensional column space can have rank at most 2, not the required rank 3. Thus the right-hand side of Theorem 3 can be nonempty while the left-hand side is empty, and the bijection in Theorem 4 fails in one direction. The statements need a spanning condition on the partial spread, or an explicit hypothesis that the union of the subspaces in the spread equals F_q^r, and the existence of such spreads must be checked for the relevant parameters. This issue affects Corollary 13, Corollary 15, and Theorem 7, which all rely on Theorem 3.","section":"Section 2, Theorem 2 and Section 3.3, Theorems 5 and 6"},{"comment":"The proof of Theorem 2 in the unequal-sublength case is omitted with the sentence 'left to the reader', but this case is exactly what is needed for the improper sum-rank Hamming code classification. Theorems 5 and 6 then state the improper analogue of the spread correspondence with 'Proofs are left to the reader' as well. Since the improper-case classification is an advertised contribution and is not reduced to previously proved results, these are load-bearing gaps. The authors should either provide complete proofs or explicitly restrict the paper's claims to the proper case until the unequal-sublength isometry classification is established.","section":"Section 3.3, Proposition 9"},{"comment":"Proposition 9 is the key characterization of codes with minimum sum-rank distance at least 3 in the m=1 case, and it is used directly in the proof of Theorem 3, yet its proof is only asserted to be 'immediate'. The statement is plausible and likely correct, but because the entire partial-spread correspondence rests on it, a full proof (or at least a detailed derivation from Theorem 1 and Lemma 6) should be included so the reader can verify both conditions and their necessity.","section":"Section 5.2, Table 1"}],"minor_comments":[{"comment":"The first row of Table 1 is internally inconsistent with the formulas in Theorem 9. For q=2, N=2, r=4, the formulas give ℓ=(2^4-1)/(2^2-1)=5, M=(N+1)ℓ=15, and k=Nℓ-r=6. The table lists ℓ=4, M=12, and k=6; the first two entries should be corrected to 5 and 15, respectively.","section":"Section 3.3, Notation"},{"comment":"In Theorems 3 and 4, the same symbol H_i denotes both a matrix block and its column space (with the latter written as H_i = Col(H_i)). This overloading makes the statement of the spread conditions harder to read; a separate notation, such as overlined or script letters for subspaces, would improve clarity.","section":"Section 3.4, Complexity remark"},{"comment":"The final complexity sentence says the term (q^r-1)/(q^N-1) is dominant over r^3+Nr and hence the running time is 'close to linear in the code length n'. This is true for fixed q and N as r grows, but it may be helpful to state the asymptotic regime explicitly, since for small r the polynomial term can be comparable.","section":"Section 1, Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has a real idea—define Hamming codes for the sum-rank metric as the longest single-error-correcting codes, and tie the m=1 case to maximal partial spreads—and the N|r case is correct. But the main classification theorem is false as stated, and there is a false lemma for m>1. Worth reading, worth fixing, not acceptable as is.\n\nWhat is new: sum-rank Hamming and simplex codes, the bijection with partial spreads (when it holds), the observation that for N|r they are perfect, the simple syndrome decoder, and the binary LRC construction. The perfect-code result (Corollary 15) is clean and the counting is right; the decoder in Section 3.4 is correct and practical; the LRC application in Theorem 9 is a nice use of binary sum-rank Hamming codes.\n\nSoft spots, roughly in increasing order of severity.\n\nFirst, Table 1's first row is wrong: for N=2, r=4, Theorem 9 gives ℓ=5, M=15, k=6, not ℓ=4, M=12. Minor typo, but it should be fixed.\n\nSecond, the abstract and introduction claim sum-rank Hamming codes are perfect for m=1 without stating the N|r condition; Corollary 15 only holds when N divides r. Overstatement.\n\nThird, several proofs are omitted. Theorem 2's extension to unequal sublengths is \"left to the reader\", and Theorems 5 and 6 are also left to the reader. These are not trivial; the improper isometry classification depends on the missing argument.\n\nFourth, Proposition 9 is stated for all m and is false for m>1. For q=2, m=2, take H_i = [1,0,ω; 0,1,ω^2] in F_4^{2×3}. Conditions 1 and 2 hold, but c=(ω,ω^2,1) has rank 2 and zero syndrome. So the characterization of d_SR≥3 via the two syndrome conditions is only valid for m=1. The paper only uses it for m=1, but the statement as written is wrong.\n\nFifth, the main issue: Theorems 3 and 4 claim a bijection between proper sum-rank Hamming codes of redundancy r and maximal-size partial N-spreads in F_q^r, but no spanning condition is imposed. The stress-test example is right: for q=2, r=3, N=2, a single plane is a maximal-size partial 2-spread, but no proper sum-rank Hamming code with redundancy 3 exists, because a parity-check matrix built from one 2D block has rank 2. The correct statement needs the partial spread to span F_q^r, which is automatic when N divides r, so Corollaries 13 and 15 survive. But the theorem as written is false.\n\nBottom line: the N|r case and the LRC application are sound and worth citing. The paper deserves a serious referee—the idea is good and the flaws are fixable—but the current version should not be taken as reliable, especially the general proper-case classification and Proposition 9. Send it to review, ask for the spanning hypothesis, the missing proofs, and a correction to Proposition 9.","headline":"A genuinely useful new class of sum-rank Hamming codes, with a real false statement in the central m=1 classification that is fixable but must be addressed before the paper is reliable.","tokens_in":18605,"tokens_out":18676,"would_cite":true,"duration_ms":189673,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B35","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines sum-rank Hamming codes as the longest single-error-correcting codes for a fixed redundancy, and proves that in the one-row case they correspond exactly to maximal-size partial spreads.","keywords":["Hamming codes","sum-rank metric","partial spreads","perfect codes","simplex codes","multishot network coding","locally repairable codes","syndrome decoding"],"falsifier":"Take $q=2$, $m=1$, with sum-rank partition $n=1+2$ and $r=3$, and search for a linear bijection on $\\mathbb{F}_q^n$ that preserves sum-rank weight but sends the 1-dimensional block into the 2-dimensional block. Such a map would violate the omitted step in Theorem 2 and would make Theorem 6's bijection between improper Hamming codes and improper partial spreads fail; a computer search over small parameters can settle this directly.","tokens_in":17519,"feed_emoji":"🧮","tokens_out":9087,"duration_ms":88039,"temperature":0.7,"pith_summary":"This paper introduces sum-rank Hamming codes, defined as the longest linear codes with minimum sum-rank distance at least 3, hence single-error-correcting, for a fixed redundancy, base-field size, and field-extension degree. The goal is to generalize classical Hamming codes to the sum-rank metric while maximizing information rate. The main structural result is that when the field-extension degree is 1, sum-rank Hamming codes are exactly the parity-check codes whose column spaces form a maximal-size partial spread, so isometry classes of such codes match maximal partial spreads one-to-one. In that setting, proper sum-rank Hamming codes are perfect when the block length divides the redundancy, and their duals, the sum-rank simplex codes, admit distance bounds derived from spread-size bounds. These codes apply to single-error correction in multishot matrix-multiplicative channels and to binary locally repairable codes.","feed_headline":"One-row sum-rank Hamming codes are maximal partial spreads","feed_subtitle":"Single-error correction in the one-row sum-rank metric becomes a packing problem, yielding perfect codes and fast decoding.","key_machinery":"The load-bearing machinery is the parity-check characterization of minimum sum-rank distance at least 3 (Proposition 9): for $m=1$ it says that a code with parity-check matrix $H=(H_1,\\dots,H_\\ell)$ has $d_{\\mathrm{SR}}\\ge 3$ exactly when the column spaces $H_i=\\operatorname{Col}(H_i)\\subseteq\\mathbb{F}_q^r$ are pairwise disjoint and each has dimension equal to its block length $n_i$. This turns code design into the geometric problem of packing disjoint subspaces, and 'longest code' becomes 'maximal-size partial spread'. The perfection argument then runs on the counting identity $|B_1|=1+\\ell(q^N-1)$ together with $\\ell=(q^r-1)/(q^N-1)$ when $N\\mid r$. The isometry classification (Theorem 2) is what upgrades an existence statement to a bijection between isometry classes of codes and of spreads.","core_discovery":"The paper's central claim is that, for $m=1$, a linear code over $\\mathbb{F}_q$ is a proper sum-rank Hamming code with redundancy $r$ and equal sublengths $N$ if and only if it admits a parity-check matrix $H=(H_1,\\dots,H_\\ell)$ whose column spaces $H_i=\\operatorname{Col}(H_i)\\subseteq\\mathbb{F}_q^r$ form a maximal-size partial $N$-spread (Theorem 3). Consequently, sum-rank isometry classes of proper sum-rank Hamming codes are in bijection with maximal-size partial $N$-spreads in $\\mathbb{F}_q^r$ (Theorem 4). When $N$ divides $r$, the number of blocks reaches $(q^r-1)/(q^N-1)$ and the codes are perfect for the sum-rank metric: the balls of radius 1 around codewords tile the entire space (Corollary 15). The same dictionary extends to unequal sublengths, now using maximal-size improper partial spreads, provided the linear isometry classification from Theorem 2 holds; the paper leaves that proof to the reader.","pith_inferences":["Inference: Because the correspondence with partial spreads is bijective, any future improvement in bounds on maximal partial spreads automatically improves the length bounds in Corollary 13 and the distance bound for sum-rank simplex codes; the paper itself uses only the simple bounds from [4] and [10].","Inference: For $m>1$, the same recipe might yield matrix-space analogues of sum-rank Hamming codes by replacing disjoint subspaces with pairwise rank-disjoint matrix spaces; the paper leaves the existence of nontrivial perfect sum-rank codes open in that regime.","Inference: The binary locally repairable code construction of Theorem 9 is flexible enough to be combined with arbitrary local linear codes, which could produce hierarchical locally repairable codes even though the paper only notes the possibility in passing."],"forward_implications":["For $m=1$ and equal sublengths $N$, the length of a proper sum-rank Hamming code with redundancy $r$ is bounded between the two expressions in Corollary 13, with equality when $N$ divides $r$.","When $N\\mid r$, proper sum-rank Hamming codes are perfect for the sum-rank metric, so the single-error-correcting capability is exactly 1 and every error of sum-rank weight at most 1 is uniquely correctable.","The syndrome decoding algorithm for $m=1$ finds the error location and value by solving linear systems against the partial-spread subspaces, with complexity $O(nr+\\ell r^3)$ over $\\mathbb{F}_q$.","The dual sum-rank simplex codes have minimum sum-rank distance at least the lower bound in Theorem 7, expressed in terms of the remainder of $r$ modulo $N$.","The codes correct a single sum-rank error in multishot matrix-multiplicative channels, and for $q=2$, $m=1$ they provide locally repairable codes with locality $N$ that can correct any pattern with one erasure per local group plus two additional erasures anywhere."],"supporting_citations":[{"why":"Supplies Theorem 1 (sum-rank distance as a minimum of Hamming distances under right block-diagonal transformations) and the Singleton-type bound used in Proposition 7.","marker":"[20]"},{"why":"Defines the sum-rank metric, the setting for all codes in the paper.","marker":"[21]"},{"why":"Introduces classical Hamming codes, the prototype that this paper generalizes to the sum-rank metric.","marker":"[11]"},{"why":"Provides the parity-check column criterion (Lemma 6) and the classical syndrome-decoding framework adapted in Section 3.4.","marker":"[13]"},{"why":"Classifies linear isometries for the rank metric, used in the proof of Theorem 2 for equal sublengths.","marker":"[2]"},{"why":"Gives the lower bound on the size of maximal-size partial spreads used in Proposition 12.","marker":"[4]"},{"why":"Gives the upper bound on the size of maximal-size partial spreads used in Proposition 12.","marker":"[10]"},{"why":"Supplies the multishot matrix-multiplicative channel model and the distance criterion (Proposition 20) used in the applications.","marker":"[19]"},{"why":"Provides the definition and basic theory of partial spreads that underpin the m=1 classification.","marker":"[3]"}],"fun_headline_variants":["Sum-rank Hamming codes reduce to partial spreads in one row","Maximal partial spreads are one-row sum-rank Hamming codes","One-row sum-rank Hamming codes are perfect and decodable","Fast decoding for sum-rank Hamming codes via partial spreads","Sum-rank Hamming codes: perfect packing in matrix channels"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification relies on the unproved claim that in a linear sum-rank isometry with unequal block sizes, a block of one size cannot be mapped to a block of a different size; if that claim fails, the correspondence between improper sum-rank Hamming codes and improper partial spreads collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sum-rank Hamming codes reduce to partial spreads in one row","Maximal partial spreads are one-row sum-rank Hamming codes","One-row sum-rank Hamming codes are perfect and decodable","Fast decoding for sum-rank Hamming codes via partial spreads","Sum-rank Hamming codes: perfect packing in matrix channels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000836,"raw_usage":{"total_tokens":3688,"prompt_tokens":1027,"completion_tokens":2661,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":2576}},"tokens_in":643,"tokens_out":2661,"duration_ms":19062,"temperature":1.0,"reasoning_tokens":2576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:21:16.174605+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $q=2$, $m=1$, with sum-rank partition $n=1+2$ and $r=3$, and search for a linear bijection on $\\mathbb{F}_q^n$ that preserves sum-rank weight but sends the 1-dimensional block into the 2-dimensional block. Such a map would violate the omitted step in Theorem 2 and would make Theorem 6's bijection between improper Hamming codes and improper partial spreads fail; a computer search over small parameters can settle this directly.","supporting_citations":[{"cited_title":"Mart ´ ınez-Pe˜ nas and F","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 1 (sum-rank distance as a minimum of Hamming distances under right block-diagonal transformations) and the Singleton-type bound used in Proposition 7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the sum-rank metric, the setting for all codes in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces classical Hamming codes, the prototype that this paper generalizes to the sum-rank metric."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parity-check column criterion (Lemma 6) and the classical syndrome-decoding framework adapted in Section 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies linear isometries for the rank metric, used in the proof of Theorem 2 for equal sublengths."},{"cited_title":"Beutelspacher","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound on the size of maximal-size partial spreads used in Proposition 12."},{"cited_title":"Gorla and A","cited_arxiv_id":null,"evidence_quote":"Gives the upper bound on the size of maximal-size partial spreads used in Proposition 12."},{"cited_title":"Mart ´ ınez-Pe˜ nas and F","cited_arxiv_id":null,"evidence_quote":"Supplies the multishot matrix-multiplicative channel model and the distance criterion (Proposition 20) used in the applications."},{"cited_title":"Beutelspacher","cited_arxiv_id":null,"evidence_quote":"Provides the definition and basic theory of partial spreads that underpin the m=1 classification."}],"review_version":1}