{"id":"965db4d3-6a20-4b9f-b02d-3d463b901239","arxiv_id":"2507.05567","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive five iterative lower bounds on error coefficients of Griesmer-optimal linear codes, prove tightness for most binary cases up to dimension 5, and determine all 5-dimensional AFER-optimal binary codes.","lead":"This paper introduces a set of iterative lower bounds for the number of minimum-weight codewords (error coefficients) in Griesmer-optimal linear codes, and uses them to identify all optimal such codes for dimensions up to 5. The bounds offer a practical way to find codes with the best asymptotic frame error rate under maximum likelihood decoding.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The remaining 5D AFER-optimality claims and the 'gap ≤ 2' assertion rest on one-line exhaustive-search assertions (Theorem 11 Case B, Theorem 13) with no reproducible code or case tables, so the central completeness claim is not yet independently verifiable.","rationale":"I read the paper in good faith and traced the main iterative arguments. Proposition 2, despite its terse proof, can be justified by an induction using the Griesmer bound on residual codes: if the punctured code had minimum distance one larger, its length would violate Griesmer, forcing the existence of the next subcode. Lemma 8's characterization of non-extendability also appears consistent with standard extensions. So the reader's weakest_assumption, while pointing at real subtleties, is not the place where the central claim is least secure. The softest point is the computational completeness that underpins the 5-dimensional classification and the 'gap ≤ 2' statement. Theorem 11's Case B and Theorem 13's proof each contain an unproduced exhaustive traversal: 'traversal puncturing' and 'iterating over all feasible solutions.' These phrases are doing essential work. Without them, we only know upper bounds from explicit constructions and lower bounds from Theorem 9 that can be weaker by 1 or 2; the claim that the listed codes are AFER-optimal, and that no other 5-dimensional cases exist, is unsupported. This is not a question of internal inconsistency but of missing reproducibility. The paper would be materially improved by providing the enumeration scripts or complete tables of residual error coefficients. Since the reader's CONDITIONAL verdict identified reproducibility concerns in its rationale, my analysis does not change the verdict; it sharpens the reason. I therefore recommend UNCHANGED, with the explicit condition that the authors supply the missing computational details.","tokens_in":27065,"tokens_out":33923,"duration_ms":328181,"concrete_test":"Reproduce the exhaustive searches. For Theorem 11, enumerate the two SS-type [40,5,20]2 codes (minihypers ℑ1 = P[4] ∪ P[3,5] and ℑ2 = P[4] ∪ P[2,4]), puncture each code in all C(40,2) pairs of coordinates, compute the minimum distance and the error coefficient of every resulting [38,5,18]2 code, and verify that the minimum error coefficient is exactly 5, with the unique (up to equivalence) minimizer matching the stated construction. For Theorem 13, similarly enumerate all puncturings of the three extended SS/Belov-type codes arising from ℑ1, ℑ2, ℑ3 and verify the claimed minima 13, 11, and 8. A short Magma or Python script, or a table of all resulting error coefficients, would settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's strongest claim is that Theorem 9's iterative bounds are tight for most binary Griesmer-optimal codes of dimension ≤ 5, and that for the remaining dimension-5 cases the gap is at most 2 (Table IV). The proof of this completeness and of the specific AFER-optimality results depends on two finite exhaustive computations that are described only in prose. In Theorem 11, Case B, the authors write: 'After traversal puncturing two [40,5,20]2 linear codes associated with ℑ1 and ℑ2, we obtain only [38,5,18;5]2 linear code. Thus, e ≥ 5.' In Theorem 13, they write: 'After iterating over all feasible solutions, we obtain e(Ĉ′1) ≥ 13, e(Ĉ′2) ≥ 11, and e(Ĉ′3) ≥ 8.' These are load-bearing: they are the only justification that the AFER-optimal error coefficient is exactly 5 (respectively 13, 11, 8) for the families [31s2+7,5,16s2+2], [31s1+8,5,16s1+3], [31s1+12,5,16s1+5], and [31s2+20,5,16s2+9]. No code, count, or detailed case table is supplied, so a reader cannot check whether the traversal was complete or whether the minimum was computed correctly. The reader's weakest_assumption focused on Proposition 2 and Lemma 8; those proofs are terse but can plausibly be completed, whereas the computational claim is an unverifiable black box. If the enumeration missed any puncturing or miscomputed any weight enumerator, the AFER-optimality assertions and the 'gap ≤ 2' conclusion could fail even though all stated inequalities are individually correct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces five iterative lower bounds on the error coefficient (number of minimum-weight codewords) of Griesmer-optimal linear codes, derived from residual codes, subcode chains, and the rank of associated minihypers. The bounds are collected in Theorem 9 and applied to binary codes of dimension up to 5. The authors claim the bounds are tight for most parameters and, for the remaining dimension-5 cases, are within 2 of the true value. They further claim AFER-optimality for several explicit families, most notably [31s2+7,5,16s2+2;5]2 (Theorem 11) and [31s1+8,5,16s1+3;13]2, [31s1+12,5,16s1+5;11]2, [31s2+20,5,16s2+9;8]2 (Theorem 13), and they provide generator matrices for the constructed codes.","tokens_in":27478,"tokens_out":13715,"duration_ms":139524,"significance":"If correct, the paper gives a practically useful method that is simpler and often tighter than the linear programming bound, and it completes the classification of binary AFER-optimal linear codes for k≤5 with explicit constructions. The paper is well organized, contains concrete examples, and supplies generator matrices in an appendix. The main caveats are that the exact AFER-optimality results for the four families in Theorem 11 and Theorem 13 rest on finite exhaustive searches that are not documented reproducibly, and one of the family-level proofs appears to check only a single parameter value. These issues affect the central completeness claim and the claimed gap of at most 2 in Table IV.","major_comments":[{"comment":"The proof of Case B only performs the puncturing traversal on the s2=1 representatives [40,5,20]2 and [38,5,18]2, but Theorem 11 claims AFER-optimality of the entire family [31s2+7,5,16s2+2;5]2 for all s2≥1. No argument shows that the conclusion for s2=1 transfers to s2≥2. Since this family is one of the four claimed AFER-optimal families in Table IV, the proof is incomplete as written.","section":"Theorem 11, Case B (Section IV-A)"},{"comment":"The assertions 'After traversal puncturing two [40,5,20]2 linear codes... we obtain only [38,5,18;5]2 linear code' and 'After iterating over all feasible solutions, we obtain e(Ĉ′1) ≥ 13, e(Ĉ′2) ≥ 11, and e(Ĉ′3) ≥ 8' are the sole justification for the exact error coefficients 5, 13, 11, and 8. No enumeration algorithm, correctness argument, case table, or code is supplied, so a reader cannot verify that the search was exhaustive or that the minima were computed correctly. Because these values underpin the 'gap ≤ 2' claim in Table IV, the manuscript should provide a reproducible script or a complete, checkable case analysis.","section":"Theorem 11, Case B, and Theorem 13 (Section IV)"},{"comment":"The 'constrains relationship' displayed in Eq. (33) is not a valid derivation as typeset: it appears to relate the ratio (Ad(C) - e(gq(k2,d),k2,q)) / e(n - gq(k2,d), k-k2, q) to ςq(k2,d)/(q-1) without proving that the punctured code Υ_A(C) has minimum distance exactly ⌈d/q^{k2}⌉ and is distance-optimal with the required parameters, or that the code attaining e(n-gq(k2,d), k-k2, q) can be realized as a subcode of C. The definitions of the divisibility corrections Δ and Δ′ are also only stated without derivation. The bound may be correct, but the proof needs to be rewritten as an explicit inequality chain.","section":"Theorem 7, Eq. (33) (Section III-D)"},{"comment":"In Case A of Proposition 2, the step claiming that an [n-d, k-1, ⌈d/q⌉+1] code 'violates the Griesmer bound' and therefore that some codeword c2 must attain n(⟨c1,c2⟩)=gq(2,d) is too quick. The existence of c2 requires an argument that if all residual codewords had weight at least ⌈d/q⌉+1, then the residual code would be an [n-d, k-1, ≥⌈d/q⌉+1] code whose parameters contradict the Griesmer bound. This can likely be completed, but as written it is a non-constructive existence claim that should be spelled out.","section":"Proposition 2 (Section III-C)"}],"minor_comments":[{"comment":"There is a typo 'theh [n − d, k− 1, ⌈d/q⌉ + 1]q linear code' that should read 'the [n − d, k− 1, ⌈d/q⌉ + 1]q linear code'.","section":"Proposition 2 proof"},{"comment":"In the row for length 31s1+11, the construction is written as '([s2·P[5]], G[42,5,20;3]2)' but the length depends on s1; the subscript should presumably be s1, and similarly in a few other rows the construction mixes s1 and s2.","section":"Table III"},{"comment":"The row for length 31s2+22 uses the symbol 'P′Tk' but the subscript k is undefined; it should be 'P′T4' or a definition should be given.","section":"Table III"},{"comment":"The use of Theorem 3 to assert e(gq(2,d),2,q)=(q-1)(t+1) when q|d is not covered by the statement of Theorem 3, since the Griesmer 2-dimensional code with length gq(2,d) is not of the form in Theorem 3 when t=0. The formula is true, but the proof should either cite a direct computation or extend Theorem 3.","section":"Theorem 4, Eq. (14)"},{"comment":"The proof of Lemma 8 is very compressed; in particular, the equivalence between extendability and the existence of a weight-d codeword c′ with wt(c+c′)=d is asserted without a full argument in both directions. A reference or a few clarifying sentences would help.","section":"Lemma 8"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the iterative-bounds idea is genuinely useful. The main barrier is reproducibility and completeness of the finite computations behind the exact error coefficients; asking the authors to provide a Magma/Python script or full case tables, and to fix the s2≥2 gap in Theorem 11, should be feasible within a revision. The proof of Theorem 7 also needs clarification. I do not see grounds for rejection, but the current evidence for the central 'gap ≤ 2' claim is not independently checkable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The iterative lower bounds in Theorems 4–8, packaged as Theorem 9, are a genuine step forward for determining AFER-optimal codes. The idea of bootstrapping error-coefficient bounds from lower-dimensional AFER-optimal codes is clean and, based on the tables, gives tight or near-tight results for most binary Griesmer-optimal codes of dimension up to 5. The paper also does something useful beyond the bounds: it completes the catalog of binary AFER-optimal codes in dimension 5, with explicit generator matrices and constructions. Lemma 8, improving Maruta's non-extendability condition for even-distance binary codes, looks like a solid standalone observation.\n\nThe soft spots are real but not evenly distributed. The proofs of Theorems 4–8 are terse; the step from (14) to (15) in Theorem 4, in particular, needs a clearer justification of the inequality and the divisibility transformations. I also found Proposition 2's existence argument non-constructive enough that I would want a more careful write-up. These are fixable in revision.\n\nThe bigger problem is the pair of exhaustive searches that carry the main completeness claims. In Theorem 11, Case B, the claim that puncturing two [40,5,20] codes yields only [38,5,18;5] is asserted in one sentence. In Theorem 13, the error coefficients 13, 11, and 8 come from \"iterating over all feasible solutions\" with no code, no case counts, and no description of the search space. These are not auxiliary checks; they are the entire justification for the exact AFER-optimality of four families and for the \"gap ≤ 2\" statement. As written, a reader cannot verify that the traversal was exhaustive or that the minimum was computed correctly. The stress-test note is right to flag this as the load-bearing weakness.\n\nThe paper is not circular and the authors are not hiding the ball—they say explicitly that they used Magma and traversal, and the appendix does provide generator matrices. But for a claim of this strength—\"we determine the parameters of all binary 5-dimensional AFER-optimal linear codes\"—one-line prose is not enough. The authors should be asked to supply the computational scripts, the exact enumeration used, and ideally a verification script that recomputes the error coefficients of the claimed optimal codes.\n\nWho is this for? Coding theorists working on error coefficients, AFER-optimality, and optimal linear codes; also people maintaining code tables. It deserves a serious referee, but the referee should demand a reproducible computational appendix before the completeness claims are accepted.","headline":"Iterative bounds are a real contribution, but the 5D completeness claims rest on black-box enumerations.","tokens_in":27970,"tokens_out":1692,"would_cite":true,"duration_ms":20704,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B65","94B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes iterative lower bounds on the number of minimum-weight codewords in Griesmer-optimal linear codes, and proves these bounds are tight for binary codes up to dimension 5 in most cases, with the remaining gap at most 2.","keywords":["error coefficient","minimum-weight codewords","AFER-optimal linear codes","Griesmer bound","iterative lower bounds","residual codes","minihypers","binary linear codes"],"falsifier":"Exhaustively enumerate all binary Griesmer-optimal \\([31s_2+7,5,16s_2+2]_2\\) codes for \\(s_2=0,1,2\\) and check whether any has an error coefficient below 5; a single code with \\(A_d=4\\) would refute Theorem 11. A broader test is to search all \\([n,5,d]\\) Griesmer-optimal codes at the lengths in Table IV and look for an error coefficient more than 2 below the Theorem 9 bound, which would refute the claimed worst-case gap.","tokens_in":26877,"feed_emoji":"📉","tokens_out":14864,"duration_ms":142193,"temperature":0.7,"pith_summary":"The paper is trying to pin down the error coefficient—the number of minimum-weight codewords—of linear codes that attain the Griesmer bound, the classical lower limit on length for a given dimension and minimum distance. Such counts matter because, on an additive white Gaussian noise channel under maximum-likelihood decoding, the smallest error coefficient among codes with the best possible distance gives the lowest asymptotic frame error rate (AFER). The paper proposes five iterative lower bounds on these coefficients, each reducing the problem to smaller-dimensional codes through residual codes, subcode chains, and projective-geometry minihypers, and packages them as a single maximum bound in Theorem 9. For binary codes of dimension at most 5 the bound is tight in most cases; the remaining 5-dimensional AFER-optimal codes are explicitly constructed, and the gap between bound and true value is at most 2. If the claims hold, a practical certificate now exists for deciding which optimal binary codes have the best asymptotic error performance.","feed_headline":"New bounds nail error coefficients of optimal binary codes","feed_subtitle":"They identify which optimal binary codes have the fewest minimum-distance error patterns, through dimension 5.","key_machinery":"The mechanism is recursion through residual codes and subcodes. For a minimum-weight codeword \\(c\\), the residual code \\(\\Upsilon_c(C)\\) has length \\(n-d\\), dimension \\(k-1\\), and minimum distance at least \\(\\lceil d/q\\rceil\\), so lower bounds for dimension \\(k-1\\) feed into \\(k\\)-dimensional bounds; Theorem 4 quantifies this via the count \\(e_{\\lceil d/q\\rceil}(n-d,k-1,q)\\). Proposition 2 supplies the structural ingredient: a Griesmer-optimal code contains a chain \\(C_1\\subset\\cdots\\subset C_{k_1}\\subset C\\) whose effective lengths equal the Griesmer values \\(g_q(i,d)\\), which Theorem 7 exploits through the difference \\(\\varsigma_q(k_2,d)\\) between consecutive Griesmer-optimal error coefficients. For codes that are exactly Griesmer, the associated minihyper \\(\\mathfrak{D}(C)\\) in projective geometry carries the counting problem, and Theorem 8 uses its rank to multiply a lower-dimensional error coefficient by \\($q^{{k-k'}}$\\). A separate tool, Lemma 8, characterizes non-extendable even-distance binary codes by the residual parameters \\([n-d,k-1,d/2]\\), powering the lower bound in Theorem 10 that handles the exceptional 5-dimensional cases.","core_discovery":"The central claim is that the error coefficient \\(A_d(C)\\) of a Griesmer-optimal \\([n,k,d]_q\\) code admits a computable lower bound obtained recursively from lower dimensions. Theorem 9 assembles five bounds \\($L_q^{{(1)}}$(n,k),\\dots,$L_q^{{(5)}}$(n,k)\\) that apply under different hypotheses—unconditionally, or when the code is Griesmer-optimal with certain values of the parameter \\(\\Gamma_q(n,k,d)\\), or when the rank of the associated minihyper is known. Iterating these bounds from the two-dimensional AFER-optimal codes of Theorem 3 fixes the error coefficients of all binary AFER-optimal codes up to dimension 4 and most of dimension 5, with explicit generator matrices or point multisets. The paper proves in Theorem 11 that \\([31s_2+7,5,16s_2+2;5]_2\\) is AFER-optimal and in Theorem 13 that \\([31s_1+8,5,16s_1+3;13]_2\\), \\([31s_1+12,5,16s_1+5;11]_2\\), and \\([31s_2+20,5,16s_2+9;8]_2\\) are AFER-optimal, completing the dimension-5 list through the two classified families of Griesmer point configurations (SS-type and Belov-type minihypers).","pith_inferences":["The same recursion should run for \\(q>2\\): the bounds in Theorems 4-8 are written for general prime powers, so constructing the analogous dimension-3/4/5 AFER-optimal databases over non-binary fields is the natural next step.","The uniform gap of at most 2 suggests that an exact formula for \\(e(n,5,2)\\) may exist; testing whether the exceptional codes form finitely many families that stabilize as \\(s\\) grows would give evidence for the paper's first open question.","Since Theorem 9's strength depends on how much is known about lower-dimensional codes, one useful stress test is to verify Proposition 2's subcode-chain existence computationally for dimension 6 before trusting the higher-dimensional bounds."],"forward_implications":["Every binary Griesmer-optimal linear code of dimension at most 5 can now be certified for AFER-optimality, with an explicit construction and a known error coefficient.","The iteration can be continued: starting from the dimension-2, 3, and 5 databases, Theorem 9 produces lower bounds for 6-dimensional Griesmer-optimal codes and, in principle, for all higher dimensions.","Codes constructed in earlier work whose AFER-optimality could not be decided by the linear programming bound are settled by the new bounds, which are also cheaper to compute.","The non-extendability criterion of Lemma 8 sharpens the existing sufficient condition and gives a template for studying extension of even-distance binary codes.","Even in the cases where the bound is not tight, the gap of at most 2 in dimension 5 implies the bound is a reliable approximation for practical code-table computations."],"supporting_citations":[{"why":"This is the Griesmer bound \\(n\\ge g_q(k,d)\\), the optimality target for all codes treated here.","marker":"[2]"},{"why":"This textbook result supplies the residual-code theorem giving \\(\\Upsilon_c(C)\\) parameters \\([n-d,k-1,\\ge\\lceil d/q\\rceil]\\), used in Theorem 4.","marker":"[19]"},{"why":"This provides the weight-enumerator and two-dimensional simplex/MDS facts behind Theorem 3's base case.","marker":"[23]"},{"why":"This gives the modified Griesmer bound used in Proposition 1 and Theorem 4.","marker":"[30]"},{"why":"This classifies binary Griesmer codes with weight-2 anti-expansion vectors as SS or Belov type, used in Lemma 4 and Section IV.","marker":"[29]"},{"why":"This defines the minihyper framework and its correspondence to Griesmer codes, used by Theorem 8 and Lemma 6.","marker":"[33]"},{"why":"This defines SS-type minihypers whose error coefficients are computed in Theorem 12.","marker":"[34]"},{"why":"This defines Belov-type minihypers and the structural lemmas used to identify the remaining 5-dimensional AFER-optimal codes.","marker":"[35]"}],"fun_headline_variants":["Iterative bounds crack error coefficients","Optimal binary codes pinned down to dimension 5","Error coefficient bounds tight to dimension 5","AFER-optimal codes complete dimension 5 picture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The iteration depends on Proposition 2's assertion that every Griesmer-optimal code contains a nested subcode chain with effective lengths exactly \\(g_q(i,d)\\); the proof finds these subcodes by a non-constructive existence argument that assumes a punctured Griesmer-violating code always contains a codeword attaining the minimum in equation (25).","fun_headline_variants_meta":{"raw":{"variants":["Iterative bounds crack error coefficients","Optimal binary codes pinned down to dimension 5","Error coefficient bounds tight to dimension 5","AFER-optimal codes complete dimension 5 picture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000742,"raw_usage":{"total_tokens":3357,"prompt_tokens":1037,"completion_tokens":2320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":653,"completion_tokens_details":{"reasoning_tokens":2263}},"tokens_in":653,"tokens_out":2320,"duration_ms":19344,"temperature":1.0,"reasoning_tokens":2263,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:21:58.823244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhaustively enumerate all binary Griesmer-optimal \\([31s_2+7,5,16s_2+2]_2\\) codes for \\(s_2=0,1,2\\) and check whether any has an error coefficient below 5; a single code with \\(A_d=4\\) would refute Theorem 11. A broader test is to search all \\([n,5,d]\\) Griesmer-optimal codes at the lengths in Table IV and look for an error coefficient more than 2 below the Theorem 9 bound, which would refute the claimed worst-case gap.","supporting_citations":[{"cited_title":"A bound for error-correcting codes,","cited_arxiv_id":null,"evidence_quote":"This is the Griesmer bound \\(n\\ge g_q(k,d)\\), the optimality target for all codes treated here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This textbook result supplies the residual-code theorem giving \\(\\Upsilon_c(C)\\) parameters \\([n-d,k-1,\\ge\\lceil d/q\\rceil]\\), used in Theorem 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This provides the weight-enumerator and two-dimensional simplex/MDS facts behind Theorem 3's base case."},{"cited_title":"Modifications of the Griesmer bound,","cited_arxiv_id":null,"evidence_quote":"This gives the modified Griesmer bound used in Proposition 1 and Theorem 4."},{"cited_title":"Further classifications of codes meeting the Griesmer bound,","cited_arxiv_id":null,"evidence_quote":"This classifies binary Griesmer codes with weight-2 anti-expansion vectors as SS or Belov type, used in Lemma 4 and Section IV."},{"cited_title":"A characterization of some [n, k, d; q]-codes meeting the Griesmer bound using a minihyper in a finite projective geometry,","cited_arxiv_id":null,"evidence_quote":"This defines the minihyper framework and its correspondence to Griesmer codes, used by Theorem 8 and Lemma 6."},{"cited_title":"Algebraically punctured cyclic codes,","cited_arxiv_id":null,"evidence_quote":"This defines SS-type minihypers whose error coefficients are computed in Theorem 12."},{"cited_title":"Construction of a class of linear binary codes achieving the Varshamov-Griesmer bound,","cited_arxiv_id":null,"evidence_quote":"This defines Belov-type minihypers and the structural lemmas used to identify the remaining 5-dimensional AFER-optimal codes."}],"review_version":1}