{"id":"20812636-9361-490a-a3b9-4feca4fd0b9b","arxiv_id":"1908.02169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Automorphism groups and equivalence criteria are determined for four families of maximum additive rank-metric codes with symmetry restrictions, and a new maximum symmetric 2-code not equivalent to the prior one is constructed.","lead":"The paper works out the full symmetry groups and equivalence classes of the main known families of maximum rank-metric codes made of symmetric, alternating, and Hermitian matrices over finite fields, and it builds a new symmetric 2-code that is provably different from the one known before.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's non-equivalence proof has an unjustified 'without loss of generality' step: product conditions allow mixed zero patterns, so the new code may still be equivalent to a known code.","rationale":"The reader's weakest_assumption (Wan's isometry classification and small exceptional parameters) does not appear to be load-bearing for the stated theorems: the exceptional symmetric q=2,n=3 codes are either the whole space (d=1) or parameter-independent (d=3), the alternating n≤3 codes are the whole alternating space, and the proof of Theorem 5.1 concerns n=2m≥4. The genuine soft spot is internal to Theorem 5.1: the unjustified WLOG step invalidates the non-equivalence proof as written. This is a correctness risk rather than a disagreement with consensus. Because the theorem may still be true and a repair may exist, the appropriate verdict remains CONDITIONAL, matching the reader's overall judgment but for a different reason.","tokens_in":16831,"tokens_out":49772,"duration_ms":482774,"concrete_test":"For m=2 (n=4), q=3 (or q=5), fix s=1 and a non-square η∈F_{q^4}^*. Use a computer algebra system to search all g(x)=g_0x+g_3x^{q^3} with -g_0/g_3∉F_q^* (so g is a permutation), together with a∈F_q^*, ρ∈Aut(F_q), and check whether the image Ψ_{a,g^T,ρ}(S) equals the known code S_{4,2,1} (i.e., its x^{q^2}-coefficient is zero and it is contained in S_4(q)). If any such equivalence exists, Theorem 5.1 is false; if exhaustive search rules this out, the gap is likely repairable by a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 5.1, after deriving the coefficient identity a_m(α)=0 for all α∈F_{q^{2m}}, the paper obtains g_{2m-i}g_{m-i}=0 for i=1,...,m. It then says 'Without loss of generality, we can suppose that g_{2m-i}=0 for i=1,...,m.' This step is not valid: the product conditions do not force all of one half to vanish. For m=2 the constraints are g_3g_1=0 and g_2g_0=0; a mixed pattern such as g_1=g_2=0 with g_0,g_3≠0 satisfies them, and g(x)=g_0x+g_3x^{q^3} can be a permutation polynomial (choose g_3 so that -g_0/g_3∉F_q^*). In such a pattern neither 'all g_{2m-i}=0' nor 'all g_{m-i}=0' holds, so the subsequent argument using c∈F_{q^m}, which assumes the upper-half coefficients g_m,...,g_{2m-1} are zero, does not apply. Since Theorem 5.1 is the basis for the claim that the constructed S is a genuinely new maximum symmetric 2-code, this is a load-bearing gap in the central assertion.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies additive maximum rank-metric codes with symmetry restrictions: symmetric, alternating, and Hermitian forms. It gives a representation of the four known families (S_{n,d,s}, A_{n,d,s}, H_{n,d,s}, E_{n,d,s}) as intersections of shifted generalized Gabidulin codes with the ambient restricted space, determines their automorphism groups, proves an equivalence criterion under which two such codes are equivalent precisely when the defining parameter s is congruent to ±s' modulo n, and characterizes the known families by a uniqueness property. It then constructs a new maximum symmetric 2-code S in S_{2m}(q) and claims, via Theorem 5.1, that this code is not equivalent to Schmidt's code S_{2m,2,s}.","tokens_in":17042,"tokens_out":24107,"duration_ms":253516,"significance":"If the results are correct, the paper provides a clean structural description of the main known restricted MRD families and settles their equivalence and automorphism classification; the new code in Section 5 would be a genuinely new object in the symmetric setting. The intersection characterization and the explicit automorphism groups are potentially useful tools for further constructions. The main novelty, however, rests on the non-equivalence proof for S, and that proof contains a load-bearing gap; the Hermitian halves of three central results are also not proved in the text. The paper is therefore interesting but needs substantial revision before its central claims are supported.","major_comments":[{"comment":"The step 'Without loss of generality, we can suppose that g_{2m-i}=0 for i=1,...,m' is unjustified. The preceding coefficient identity gives only g_{2m-i}g_{m-i}=0 for each i, i.e. at least one factor in each antipodal pair vanishes. Mixed zero patterns are allowed. For example, when m=2, the conditions are g_3g_1=0 and g_2g_0=0; choosing g_1=g_2=0 and g_0,g_3 nonzero satisfies them, and g(x)=g_0x+g_3x^{q^{3s}} can be a permutation polynomial for suitable g_3. In such a pattern neither all upper-half coefficients nor all lower-half coefficients vanish, so the later argument that uses c∈F_{q^m} and concludes g_i=0 for i=0,...,m-1 does not apply. Since Theorem 5.1 is the only evidence that the constructed code S is new, this gap is load-bearing for the paper's main claim.","section":"§5, Theorem 5.1"},{"comment":"The Hermitian cases are not proved. Theorem 3.3(ii) is dismissed with 'the proof of this point is similar to that of previous one. For this reason we omit here the computations' and Theorem 4.2 with the same phrase. The Hermitian half of Theorem 3.5 is also delegated to 'similar arguments'. Since these theorems are stated as full classifications for H_{n,d,s} and E_{n,d,s}, the analogous coefficient arguments should be supplied or at least reduced in detail to the symmetric case; as written, the Hermitian claims are unsupported.","section":"§3, Theorem 3.3(ii); §3, Theorem 3.5; §4, Theorem 4.2"},{"comment":"The paper invokes Wan's classification of rank-preserving maps with stated exceptions (q=2,n=3 for S_n(q); n≤3 for A_n(q)), but the subsequent theorem statements do not exclude those parameters. In particular, Corollary 3.4 and Theorem 3.5 cover A_{n,d,s} for all n with 1<d<n, including n=3 and d=2, where the exceptional isometries of A_n(q) could alter the automorphism group or create additional equivalences. The authors should either exclude the exceptional parameters explicitly or verify that the theorems remain true in those cases.","section":"§2 (after Eq. (9)) and §3, Corollary 3.4 / Theorem 3.5"}],"minor_comments":[{"comment":"The condition on s is printed as 'gcd(s,2m)' without the required equality '=1'; please correct.","section":"§5, first paragraph"},{"comment":"In the Hermitian part, the hypotheses use gcd(s,2n)=gcd(s',2n)=1 but the conclusion is s≡±s' (mod n). Since the Hermitian codes depend on s only modulo n, this is consistent, but the mismatch between the modulus in the hypothesis and the conclusion should be clarified.","section":"§3, Theorem 3.5"},{"comment":"The sentence 'the coefficients of terms x and x^{q^{s(2m-1)}} of f_m are c and ηb' uses an undefined symbol c; it should presumably be a (or another named coefficient).","section":"§5, paragraph before Theorem 5.1"},{"comment":"The index conventions in the displayed formula for c_{m,j}(α_j) are hard to reconcile with the composition g∘αx^{q^{sj}}∘h; the power of q on h_{m-i-j} appears to be q^{s i} rather than q^{s(i+j)}. Please check the displayed formula and the resulting condition (23), and spell out the index arithmetic that yields the set equality after (23).","section":"§3, proof of Theorem 3.3"},{"comment":"The displayed bound for alternating codes is ambiguous: 'q^{n(n−1) 2m (m−e+1)}' should be written with the division by 2m clearly indicated.","section":"§2, Theorem 2.3"}],"recommendation":"major_revision","confidential_remarks":"The new construction in Section 5 is potentially the most valuable part of the paper, but the proof of its non-equivalence is currently incomplete at the WLOG step. The omitted Hermitian arguments and the unhandled Wan exceptions are secondary but should be fixed before the paper can be considered. I do not see evidence of circularity or fitted parameters; the difficulties are in the proof details, not in the framing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper is worth your attention: the automorphism-group and equivalence results for the four restricted MRD families are genuinely new and mostly well argued, but the advertised new symmetric 2-code in Section 5 is not yet proved to be new.\n\nWhat the paper does well. Proposition 3.1 gives a clean characterization: each restricted maximum code is the intersection of its ambient symmetric/alternating/Hermitian space with a suitable shifted Gabidulin code. Theorem 3.3 shows that this intersection is unique, and the consequences—the automorphism groups in Corollary 3.4 and the equivalence criterion s ≡ ±s′ (mod n) in Theorem 3.5—are strong and appear correct for the symmetric and alternating cases, which are proved in detail. The Hermitian versions are only sketched, but the pattern is clear. Theorem 4.1's characterization of equivalence via a unique self-adjoint ambient subspace is also a nice contribution.\n\nThe soft spots. The proof of Theorem 5.1, which claims the new code S is not equivalent to S_{2m,2,s}, contains an invalid 'without loss of generality' step. From the coefficient vanishing the authors get g_{2m-i} g_{m-i}=0 for each i=1,...,m, but this only says that for each residue class modulo m at most one coefficient from the pair can be nonzero. It does not force all upper-half or all lower-half coefficients to vanish. Mixed patterns are possible, and some mixed patterns can even give permutation polynomials. The subsequent argument using c ∈ F_{q^m} depends on assuming the upper half is zero, so the non-equivalence proof does not go through as written. This is a load-bearing gap: the new-code claim is the headline of Section 5.\n\nA second, smaller issue: the authors rely on Wan's classification of rank-preserving maps, which has small exceptional parameters, but the theorem statements do not exclude them. For example, q=2, n=3 for the symmetric case and n≤3 for alternating. This is likely fixable, but as stated the theorems are overbroad.\n\nBottom line. The intersection characterization and automorphism-group results are substantial and probably correct. The new 2-code may well be new, but the proof given is incomplete. The paper deserves a serious referee; I'd send it out with a specific request to verify Theorem 5.1. If that claim is repaired or removed, the rest is publishable.","headline":"Strong automorphism-group results for restricted MRD codes, but the non-equivalence proof for the new symmetric 2-code rests on an invalid WLOG step.","tokens_in":17622,"tokens_out":6731,"would_cite":false,"duration_ms":100168,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows the four known optimal restricted rank-metric code families are exactly intersections with shifted Gabidulin codes, yielding their automorphism groups, the equivalence criterion, and a new symmetric 2-code.","keywords":["rank metric codes","MRD codes","generalized Gabidulin codes","symmetric bilinear forms","alternating bilinear forms","Hermitian forms","automorphism groups","code equivalence"],"falsifier":"The weakest point is easy to probe: for the excluded small parameters ($q=2,n=3$ for the symmetric space, $n\\le 3$ for the alternating space), exhaustively compute the automorphism group of $S_{n,d,s}$ or $A_{n,d,s}$ over all rank-preserving maps; a single extra map would falsify Corollary 3.4 and Theorem 3.5. For the new code, a direct search over all maps $\\Psi_{a,g,\\rho}$ for small $m,q$ sending the Section 5 code into $S_{2m,2,s}$ would settle Theorem 5.1.","tokens_in":16584,"feed_emoji":"🧮","tokens_out":13385,"duration_ms":113238,"temperature":0.7,"pith_summary":"The paper studies maximum additive rank-metric codes whose matrices are required to be symmetric, alternating, or Hermitian. It establishes that the four known families of such codes are not isolated examples: each family is exactly the intersection of its restricted matrix space with a suitably shifted generalized Gabidulin code, and the shifted Gabidulin space is uniquely determined. From that intersection description it derives the full automorphism group of each family and proves that two codes are equivalent precisely when their defining shift exponents satisfy the same congruence up to sign. It also constructs a new maximum symmetric 2-code in even dimension with the same size as the known family but provably not equivalent to it. The result is a complete equivalence classification of the known restricted optimal codes and evidence that the list of such codes is not yet complete.","feed_headline":"Optimal restricted rank codes: one shift decides equivalence","feed_subtitle":"Optimal symmetric, alternating, and Hermitian codes are Gabidulin intersections; only s and −s are equivalent.","key_machinery":"The load-bearing object is the generalized Gabidulin code $G_{n,k,s}$, the set of linearized polynomials with q-degrees confined to $0,s,2s,\\dots,(k-1)s$, together with its right-composition shifts $G\\circ x^{q^{st}}$. The paper's key identity realizes each restricted code as $G'\\cap X_n$, where $X_n$ is the subspace of self-adjoint symmetric, alternating, or Hermitian maps. What makes the identity powerful is Theorem 3.3: among all equivalent copies of the ambient Gabidulin code, the shifted container $G'$ is the unique one meeting $X_n$ in the given restricted code. Combined with the known classification of rank-preserving maps on these restricted spaces (all isometries have the form $\\Psi_{a,g,\\rho,r_0}(f)=a\\,g\\circ f^\\rho\\circ g^\\top+r_0$, with an analogous $\\Theta$ in the Hermitian case) and with the known automorphism group of $G_{n,k,s}$, the argument reduces automorphisms and equivalence of the restricted codes to monomial maps and to the congruence of $s$ modulo $n$.","core_discovery":"The paper's central discovery is the intersection identity of Proposition 3.1: in q-polynomial form, the four known restricted maximum codes can be written as $S_{n,d,s}=G_{n,n-d+1,s}\\circ x^{q^{s(n+d)/2}}\\cap S_n(q)$, $A_{n,d,s}=G_{n,n-d+1,s}\\circ x^{q^{sd/2}}\\cap A_n(q)$, $H_{n,d,s}=G_{n,n-d+1,s}\\circ x^{q^{s(n+d+1)}}\\cap H_n(q^2)$, and $E_{n,d,s}=G_{n,n-d+1,s}\\circ x^{q^{s(d+1)}}\\cap H_n(q^2)$, where $G_{n,k,s}$ is the generalized Gabidulin code with minimum distance $d=n-k+1$. Theorem 3.3 proves that the shifted Gabidulin container is unique, and Theorem 4.1 characterizes every code equivalent to these families as the unique self-adjoint intersection $D=V\\cap X_n$ with $V=V^\\top$. From this the paper derives the automorphism groups in Corollary 3.4, proves the equivalence criterion $s\\equiv\\pm s'\\pmod n$ in Theorem 3.5, and exhibits in Theorem 5.1 a maximum symmetric 2-code that is not equivalent to $S_{2m,2,s}$.","pith_inferences":["If the uniqueness characterization extends to the punctured symmetric codes $T_{n,d,s}(\\eta)$, those codes should also admit a Gabidulin-container description; checking that would give a direct route to their automorphism groups.","The same coefficient-comparison technique used against the new symmetric 2-code could be used as a test for equivalence between that code and any future symmetric 2-code family.","A natural classification question left implicit by the paper is whether the maximum symmetric 2-codes in $S_{2m}(q)$ consist of exactly two inequivalent families; the two examples here make that a concrete finite search."],"forward_implications":["Each of the four known families has exactly the stated monomial automorphism group; no rank-preserving map outside the displayed list fixes the code.","Within each family, the congruence $s\\equiv\\pm s'\\pmod n$ is a complete invariant for equivalence: codes built from $s$ and $-s$ are equivalent, while codes built from inequivalent residues are not.","Every maximum code equivalent to one of these families is the intersection of its restricted space with a uniquely determined self-adjoint shifted Gabidulin space, so the whole equivalence class has the same structural description.","The new Section 5 code is a maximum symmetric 2-code with the same parameters as $S_{2m,2,s}$ but not equivalent to it, so the known symmetric family does not exhaust all maximum symmetric 2-codes."],"supporting_citations":[{"why":"Supplies the classification of rank-preserving maps on symmetric, alternating, and Hermitian spaces that defines equivalence in restricted spaces and is used throughout.","marker":"[19]"},{"why":"Provides the generalized Gabidulin codes $G_{n,k,s}$ that serve as the container in the intersection identities.","marker":"[7]"},{"why":"Constructs the symmetric codes $S_{n,d,s}$ and proves their maximality, the objects whose automorphisms and equivalence are determined.","marker":"[16]"},{"why":"Constructs the alternating codes $A_{n,d,s}$ and proves the bound they attain.","marker":"[2]"},{"why":"Constructs the Hermitian codes $H_{n,d,s}$ and $E_{n,d,s}$.","marker":"[13]"},{"why":"Supplies the automorphism group of generalized Gabidulin codes used to derive Corollary 3.4.","marker":"[17]"},{"why":"Supplies the equivalence theorem for generalized Gabidulin codes used in Theorem 3.5 to translate $s\\equiv\\pm s'$ into an explicit map.","marker":"[8]"},{"why":"Supplies the sharp upper bound for symmetric codes used to rule out a larger intersection in the automorphism-group proof.","marker":"[14]"},{"why":"Supplies the MRD code $D_{2m-1,s}(\\eta)$ used as the container for the new symmetric 2-code in Section 5.","marker":"[18]"},{"why":"Supplies the rank lower bound used to show the new Section 5 code has minimum distance at least 2.","marker":"[5]"}],"fun_headline_variants":["Shift sign pins down equivalence in restricted rank codes","New symmetric 2-code escapes known equivalence class","Automorphism groups of optimal restricted rank codes solved","Intersection identity settles equivalence for Gabidulin codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the full classification of rank-preserving maps on the restricted symmetric, alternating, and Hermitian matrix spaces; if that classification misses exceptional cases, then the automorphism groups and the equivalence criterion would have extra cases beyond the ones stated.","fun_headline_variants_meta":{"raw":{"variants":["Shift sign pins down equivalence in restricted rank codes","New symmetric 2-code escapes known equivalence class","Automorphism groups of optimal restricted rank codes solved","Intersection identity settles equivalence for Gabidulin codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1607,"prompt_tokens":1036,"completion_tokens":571,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":511}},"tokens_in":652,"tokens_out":571,"duration_ms":5912,"temperature":1.0,"reasoning_tokens":511,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:52:32.872345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The weakest point is easy to probe: for the excluded small parameters ($q=2,n=3$ for the symmetric space, $n\\le 3$ for the alternating space), exhaustively compute the automorphism group of $S_{n,d,s}$ or $A_{n,d,s}$ over all rank-preserving maps; a single extra map would falsify Corollary 3.4 and Theorem 3.5. For the new code, a direct search over all maps $\\Psi_{a,g,\\rho}$ for small $m,q$ sending the Section 5 code into $S_{2m,2,s}$ would settle Theorem 5.1.","supporting_citations":[{"cited_title":"W an: Geometry of matrices , W orld Scientific, Singapore, 1996","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of rank-preserving maps on symmetric, alternating, and Hermitian spaces that defines equivalence in restricted spaces and is used throughout."},{"cited_title":"Gabidulin, A","cited_arxiv_id":null,"evidence_quote":"Provides the generalized Gabidulin codes $G_{n,k,s}$ that serve as the container in the intersection identities."},{"cited_title":"Schmidt: Symmetric bilinear forms over finite field s with applications to coding theory, Journal of Algebraic Combinatorics , vol","cited_arxiv_id":null,"evidence_quote":"Constructs the symmetric codes $S_{n,d,s}$ and proves their maximality, the objects whose automorphisms and equivalence are determined."},{"cited_title":"Delsarte, J.M","cited_arxiv_id":null,"evidence_quote":"Constructs the alternating codes $A_{n,d,s}$ and proves the bound they attain."},{"cited_title":"Schmidt: Hermitian rank distance codes, Designs, Codes and Cryptography , 86(7), p","cited_arxiv_id":null,"evidence_quote":"Constructs the Hermitian codes $H_{n,d,s}$ and $E_{n,d,s}$."},{"cited_title":"Sheekey: A new family of linear maximum rank distance codes, Advances in Mathematics of Communications , 2016","cited_arxiv_id":null,"evidence_quote":"Supplies the automorphism group of generalized Gabidulin codes used to derive Corollary 3.4."},{"cited_title":"Lunardon, R","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence theorem for generalized Gabidulin codes used in Theorem 3.5 to translate $s\\equiv\\pm s'$ into an explicit map."},{"cited_title":"Quadratic and symmetric bilinear forms over finite fields and their association schemes","cited_arxiv_id":"1803.04274","evidence_quote":"Supplies the sharp upper bound for symmetric codes used to rule out a larger intersection in the automorphism-group proof."},{"cited_title":"Trombetti, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the MRD code $D_{2m-1,s}(\\eta)$ used as the container for the new symmetric 2-code in Section 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the rank lower bound used to show the new Section 5 code has minimum distance at least 2."}],"review_version":1}