{"id":"f3f37b79-e395-45b6-8403-9f7a04a4a138","arxiv_id":"1908.03294","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using a simplex-code correspondence, the paper establishes exact optimal minimum weights and classifications for binary LCD codes of dimensions 3 and 4 and ternary LCD codes of dimensions 2 and 3.","lead":"The authors determine the best possible minimum distances and complete optimal-code lists for small binary and ternary LCD codes. A new reduction theorem turns an infinite classification problem into finite computer checks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Arbitrary-n results rest on unshipped exhaustive-search claims; a missed m-vector or equivalence mis-grouping at any base length would invalidate all propagated classifications.","rationale":"The reader's weakest assumption correctly identifies the exhaustive computer searches in Section 5 as the load-bearing step. My stress-test found no internal mathematical flaw in Theorem 4.5 or in the reduction arguments: the construction of C0, the rank argument implying q*r >= k, the constant-weight/self-orthogonal concatenation in Lemma 3.5, and the equivalence lemmas all check out. The one genuinely insecure point is that the arbitrary-n theorems inherit the correctness of finite base classifications and nonexistence statements that are asserted solely on the basis of unshipped C programs and nauty/Magma runs. The reported data for representative codes does not provide a negative certificate, so a missed m-vector would silently propagate through Theorem 4.7 to infinitely many lengths. This concern is substantial but does not by itself demonstrate an error; it calls for independent verification or an archived search certificate. Since the reader already assigned CONDITIONAL for exactly this reason, my verdict is unchanged.","tokens_in":23437,"tokens_out":20333,"duration_ms":205029,"concrete_test":"Independently re-implement the Section 5 enumeration for one propagated nonexistence base case, e.g. ternary LCD [27,3,18] (r=9): enumerate every m=(m1,...,m13) in Z_{\\ge 0}^{13} with sum 27 and qd-(q-1)n <= m_i <= n - ((q^{k-1}-1)/((q-1)q^{k-2}))d, check GGT nonsingularity and exact minimum weight, and verify that no LCD code exists. In the same harness recompute a positive base count such as N3(51,3)=15 and archive the code and certificate; if the independent search reproduces both outcomes, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reduction in Theorem 4.5 is carefully argued and the rank/Griesmer bookkeeping checks out. The load-bearing point is the finite computational base: Theorem 4.7 propagates classification and nonexistence from short [2r,4,r] and [3r,3,2r] (and related) lengths to every n in the residue class. Those base results come entirely from the Section 5 enumeration of m-vectors satisfying (9), with no program source and no certificate that the enumeration was exhaustive. The text reports NTL determinant checks for LCD membership and nauty for equivalence, and says Magma verified the reported codes, but verifying the codes that were found does not certify that no other m-vector was missed. A single missed vector, or an incorrect graph-isomorphism equivalence decision, would change a base count or turn a claimed nonexistence into an existence, and Theorem 4.7 would then propagate the error to all s. Thus the central classification and d_q values are only as secure as the finite searches, whose completeness is not independently checkable from the paper. This is a correctness risk, not merely a reproducibility inconvenience: the infinite families inherit every undetected error in the unshipped search programs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies LCD codes over F2 and F3. The main theoretical result, Theorem 4.5, establishes a one-to-one correspondence between equivalence classes of LCD [n,k,d] codes with dual distance at least 2 and equivalence classes of LCD [q r, k, (q-1) r] codes with dual distance at least 2, where r = q^{k-1} n - [k]_q d, under the assumptions (q,k0) in {(2,3),(3,2)}, k >= k0, qd-(q-1)n >= 1, and q r >= k. This reduces classification and nonexistence for infinite families of lengths to finite base cases. Using Theorem 4.7 (a reformulation adapted to the residue-class structure of the lengths) and exhaustive computer searches over the m-vectors satisfying (9), the paper determines the largest minimum weights d_q(n,k) for (q,k) in {(2,4),(3,2),(3,3)} and gives classifications of optimal LCD codes for (q,k) in {(2,3),(2,4),(3,2),(3,3)} for all n.","tokens_in":80,"tokens_out":21124,"duration_ms":311094,"significance":"If the computational base cases are correct, the paper settles the optimal-LCD classification problem for the four small-dimension families over F2 and F3, a genuinely useful contribution given the recent interest in LCD codes. The theoretical reduction in Theorem 4.5 is elegant and parameter-free: it derives the correspondence from the structure of simplex codes and symmetric designs (Lemma 3.6) and avoids relying on previously conjectured bounds. The proof of the correspondence is checkable and the algebra leading to r and (q-1)r is clean. The classification tables and the explicit lists of optimal codes are valuable data. The main risk is that the arbitrary-n results rest on an exhaustive-search foundation that is not independently verifiable from the manuscript; this is a correctness risk rather than a mere reproducibility inconvenience, because Theorem 4.7 propagates every base-case count or nonexistence to all lengths in the residue class.","major_comments":[{"comment":"The exhaustive enumeration over all m-vectors satisfying (9) is the load-bearing computational premise for the arbitrary-n results. The text reports that all such vectors were considered and that NTL determinant checks and nauty equivalence tests were used, and that Magma verified that the reported codes are LCD and inequivalent; but no search program, search log, or certificate of completeness is provided. Magma verification of the codes that were found does not certify that no other m-vector was missed, and a single missed vector at a base length (for example, one that would yield a binary [16,4,8] or [30,4,15] code, or a ternary [3r,3,2r] code for r in {4,6,7,8,9}) would change a classification count or a nonexistence statement that Theorem 4.7 then propagates to every length in the corresponding residue class. Please make the search programs available or supply a machine-checkable certificate (or an independent full computation) that the enumerations in Propositions 6.1, 6.4, 6.6, 7.4, 7.7, and 7.9 are complete.","section":"Section 5, Theorem 4.7"},{"comment":"The paper claims a complete classification of optimal LCD codes, but the stated counts and the tables N_q(n,k) refer only to codes with dual distance at least 2. Since d_q(n,k)=d_q(n-1,k) for several residue classes (for example, d2(7s+1,3)=d2(7s,3) and d2(15s,4)=d2(15s-1,4)), zero-padding every optimal code of length n-1 yields additional optimal codes of length n with dual distance 1, and these are not included in the listed numbers. Lemma 3.1 is invoked, but the paper should state explicitly how these padded classes are part of the classification and, if the classification is meant to enumerate all equivalence classes, provide the total counts or an explicit recursive description of the full set of optimal codes for each length.","section":"Propositions 6.2, 6.7, 7.5, 7.10 (classification statements)"}],"minor_comments":[{"comment":"The statement says 'ternary LCD [3r,2,2r] codes' but the dimension should be 3, as is clear from the context and from the table of parameters in the same subsection.","section":"Proposition 7.9"},{"comment":"The proof that the constructed short code C0 has dual distance at least 2 is omitted. This holds because the columns of G0 are a subset of the columns of G and C has no zero columns, but the argument should be stated explicitly given that Theorem 4.5(i) includes the condition d^⊥ >= 2 on both sides of the correspondence.","section":"Lemma 4.1 / Theorem 4.5"},{"comment":"The canonical condition m1,m2,m4 >= 1 (and the analogous conditions for the other cases) should be justified in one sentence: these indices correspond to the unit coordinate columns in the recursive ordering of the simplex matrix S_{q,k}, so any code with dual distance at least 2 can be brought to this form.","section":"Section 5"},{"comment":"The definition of s' via the fraction (qr - t)/[k]_q + 1 is less transparent than the equivalent identity qr = [k]_q(s' - 1) + t used in the proof of Theorem 4.7; introducing the latter earlier would improve readability.","section":"Equation (16)"}],"recommendation":"major_revision","confidential_remarks":"The base tables and several nonexistence results in Sections 6-7 are taken from the authors' earlier papers ([1], [2], [12], [13]); this is not circular because Theorem 4.5 does not invoke those results, but the arbitrary-n conclusions inherit any errors in those tables as well as in the new Section 5 searches. Given the centrality of the unshipped exhaustive search, I would urge the editor to require the authors to deposit the search programs or a machine-checkable certificate before publication. The paper fits the journal's scope well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuinely useful paper, and I'd send it out. The main theorem is a real reduction, not just a table dump, and the tables and classifications are a service to people working on binary and ternary LCD codes. The one thing that keeps me from treating the infinite-family results as settled is that they inherit from finite computer searches whose completeness is not independently verifiable from the manuscript.\n\nThe new content: Theorem 4.5 gives a one-to-one correspondence between equivalence classes of LCD [n,k,d] codes with dual distance at least 2 and LCD [q r, k, (q-1)r] codes with dual distance at least 2, under simple hypotheses, with r = q^{k-1}n - [k]_q d. The proof is coherent: Lemma 3.6 packages the simplex-code/design inequality, and Lemmas 4.3 and 4.4 provide both directions of the bijection. Theorem 4.7 converts the reduction into a periodic propagation statement, which is exactly why the classification stabilizes in n. The resulting determination of d_2(n,4), d_3(n,2), d_3(n,3) and the classifications for (q,k) in {(2,3), (2,4), (3,2), (3,3)} complete the small-dimension picture. The authors also correctly note that [20] independently determined d_3(n,2). Self-citation to earlier classifications is disclosed and is not circular.\n\nSoft spots. The computational base cases in Section 5 are load-bearing. Section 5 reports NTL determinant checks and nauty isomorphism testing, and says Magma verified the reported codes, but the search programs and exhaustive certificates are not shipped. A missed m-vector or a wrong equivalence decision at a base length would propagate through Theorem 4.7 to every length in the residue class. That is a correctness risk, not merely a reproducibility inconvenience, so I would ask the authors to archive the C programs and enough certificates before final acceptance. Minor issues: Proposition 7.9 has a dimension typo, saying [3r,2,2r] where it should say [3r,3,2r]; and in Lemma 4.1 the dual-distance condition on the short code is implicit rather than stated. Neither affects the argument.\n\nWho should read this: coding theorists working on LCD codes, especially people interested in side-channel countermeasures or EAQECCs from binary and ternary codes. The tables are genuinely useful reference data. With code and certificates archived, I would accept; without them, the classification claims are best described as conditional. Either way, the paper deserves a serious referee.","headline":"Solid small-table completion with a real structural reduction; the only serious caveat is that the arbitrary-n claims lean on unshipped exhaustive searches.","tokens_in":24204,"tokens_out":4893,"would_cite":true,"duration_ms":48168,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"A simplex-code identity classifies all optimal binary and ternary LCD codes","keywords":["LCD codes","linear complementary dual codes","minimum weight","optimal codes","simplex codes","dual distance","Griesmer bound","code classification"],"falsifier":"Run an independent exhaustive search for a binary LCD [16,4,8] code with dual distance at least 2: the paper's claimed value d_2(16,4)=7 says no such code exists, so finding one would break the nonexistence propagation chain for the n=15s+1 family. Equivalently, enumerate all multiplicity vectors m satisfying the bounds (9) for (q,k,n,d)=(2,4,16,8) and check the LCD condition; any survivor is a direct counterexample.","tokens_in":23222,"feed_emoji":"🔢","tokens_out":6989,"duration_ms":70874,"temperature":0.7,"pith_summary":"The paper establishes a structural correspondence for linear complementary dual (LCD) codes over the binary and ternary fields with large minimum weight: every such code is built from a self-orthogonal simplex block plus a shorter LCD tail, and this decomposition is unique up to equivalence. Carried by the simplex code's columns, the correspondence maps an LCD [n,k,d] code with dual distance at least 2 to an LCD [q*r, k, (q-1)r] code, where r = $q^{{k-1}}$n - [k]_q d. Because the map is a bijection on equivalence classes, nonexistence and classifications proved for small lengths propagate to every length in the same residue family. The paper uses this to determine, for arbitrary n, the largest minimum weights d_2(n,4), d_3(n,2), d_3(n,3) and to classify all optimal LCD codes for (q,k) equal to (2,3), (2,4), (3,2), and (3,3).","feed_headline":"Optimal LCD codes fully classified for binary and ternary, all lengths","feed_subtitle":"A simplex-code identity turns infinite classification into finite checks, settling minimum-weight formulas for every n.","key_machinery":"The carrying object is the simplex code S_{q,k}: a generator matrix whose columns are the nonzero projective vectors, arranged so the code is self-orthogonal and constant-weight, with length [k]_q = (q^k - 1)/(q - 1) and minimum weight $q^{{k-1}}$. Any LCD code with dual distance at least 2 is equivalent to a code C_{q,k}(m) built by repeating these columns according to multiplicities m_i. Lemma 3.6 bounds each multiplicity using the symmetric 2-design formed by the simplex supports, giving qd - (q-1)n <= m_i and an upper bound in terms of d. Lemma 3.5 shows that adjoining s copies of the simplex block raises the length by [k]_q s and the minimum weight by $q^{{k-1}}$s while preserving the LCD property. Together these produce the bijection of Theorem 4.5, so that classifying base codes of length q*r automatically classifies all codes of length [k]_q s + t in the family.","core_discovery":"The central discovery is Theorem 4.5: under the conditions qd - (q-1)n >= 1, k >= k0 with (q,k0) in {(2,3), (3,2)}, and q*r >= k, where r = $q^{{k-1}}$n - [k]_q d, there is a one-to-one correspondence between equivalence classes of LCD [n,k,d] codes over F_q with dual distance at least 2 and equivalence classes of LCD [q*r, k, (q-1)r] codes with dual distance at least 2. In the modified form of Theorem 4.7, writing n = [k]_q s + t and d = $q^{{k-1}}$s + $\\alpha$(t), the classification of optimal codes for all lengths in that residue class is exactly the classification of the finite base code with parameters [q*r, k, (q-1)r], and nonexistence at the base length rules out the whole infinite family. The paper applies this to prove explicit formulas for d_2(n,4), d_3(n,2), and d_3(n,3), and to give a complete classification of optimal LCD codes for the four listed parameter families, with counts of inequivalent codes that are eventually periodic in the length n.","pith_inferences":["Editorial inference: the same 'self-orthogonal constant-weight block plus LCD tail' decomposition could be adapted to other code classes that contain a self-orthogonal constant-weight subcode, potentially yielding analogous propagation of classification results for near-MDS codes or codes with prescribed dual distances.","Editorial inference: the periodic structure implies that the asymptotic number of inequivalent optimal LCD codes for fixed q and k is governed by finitely many constants, so the classification problem does not grow harder with n once the base lengths are resolved.","Editorial inference: providing machine-checkable certificates, or publishing the exhaustive-search programs, for the base-length nonexistence statements would convert the computational part of the proof into independently verifiable evidence; the present paper states the search results but does not ship the code."],"forward_implications":["For every length n, the largest minimum weights d_2(n,4), d_3(n,2), and d_3(n,3) are determined by explicit formulas depending only on n modulo 15, 4, and 13 respectively, e.g., d_2(n,4) = floor(8n/15) for n congruent to 5, 9, or 13 mod 15, and floor(8n/15) - 2 for n divisible by 15.","The complete classification of optimal LCD codes for (q,k) in {(2,3), (2,4), (3,2), (3,3)} is finite in each residue class: for each residue of n modulo [k]_q, the number of inequivalent optimal codes with dual distance at least 2 becomes constant for all large enough n, with the paper giving the exact counts.","Because the correspondence is a bijection on equivalence classes, any future independent classification of the short base lengths [q*r, k, (q-1)r] automatically upgrades to a full classification for all lengths in the corresponding residue family without any new search.","The nonexistence part of the theorem rules out Griesmer-bound-attaining LCD codes for infinite families, such as binary LCD [15s, 4, 8s] codes and ternary LCD [13s, 3, 9s] codes for every positive integer s.","The classification data from small lengths is enough to certify optimality for arbitrarily large n: the paper's results settle d_q(n,k) and the number of optimal codes for every n in the covered families, not just up to some finite cutoff."],"supporting_citations":[{"why":"Supplies the prior classification of binary and ternary LCD codes of small lengths, which serves as the base data for the propagation argument.","marker":"[1]"},{"why":"Provides the design-based inequality used to bound the column multiplicities in Lemma 3.6.","marker":"[3]"},{"why":"Gives the structural fact that equidistant linear codes are sequences of dual Hamming codes, used to handle lengths divisible by [k]_q.","marker":"[4]"},{"why":"Supplies characterizations and generator-matrix forms for LCD codes that underlie the component constructions in Lemmas 3.2 and 3.5.","marker":"[8]"},{"why":"Establishes the earlier determination of d_2(n,3) and a partial classification of binary optimal LCD [n,3] codes that the present paper completes.","marker":"[12]"},{"why":"Provides the standard facts about simplex codes, self-orthogonality, constant weight, and the symmetric design structure used in Lemma 3.6.","marker":"[14]"},{"why":"Introduces LCD codes and proves the generator-matrix criterion G G^T nonsingular used throughout to test the LCD property.","marker":"[18]"},{"why":"Supplies graph-isomorphism testing used to decide equivalence among the codes found in the exhaustive searches.","marker":"[19]"},{"why":"Supplies the determinant computation routine used to check nonsingularity of G G^T in the search program.","marker":"[21]"}],"fun_headline_variants":["Finite checks classify infinite family of optimal LCD codes","All optimal LCD codes for q=2,3 now classified","Infinite LCD code classification reduced to finite base cases","Complete classification of optimal LCD codes over F2 and F3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the exhaustive computer searches at the small base lengths found every relevant code and counted equivalence classes correctly; a single missed code at a base length would invalidate the conclusions for infinite families, and the search programs are not shipped.","fun_headline_variants_meta":{"raw":{"variants":["Finite checks classify infinite family of optimal LCD codes","All optimal LCD codes for q=2,3 now classified","Infinite LCD code classification reduced to finite base cases","Complete classification of optimal LCD codes over F2 and F3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2681,"prompt_tokens":922,"completion_tokens":1759,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":1692}},"tokens_in":538,"tokens_out":1759,"duration_ms":11359,"temperature":1.0,"reasoning_tokens":1692,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:18:53.940548+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent exhaustive search for a binary LCD [16,4,8] code with dual distance at least 2: the paper's claimed value d_2(16,4)=7 says no such code exists, so finding one would break the nonexistence propagation chain for the n=15s+1 family. Equivalently, enumerate all multiplicity vectors m satisfying the bounds (9) for (q,k,n,d)=(2,4,16,8) and check the LCD condition; any survivor is a direct counterexample.","supporting_citations":[{"cited_title":"Araya, M","cited_arxiv_id":null,"evidence_quote":"Provides the design-based inequality used to bound the column multiplicities in Lemma 3.6."},{"cited_title":"Bonisoli, Every equidistant linear code is a sequence of dual Ham ming codes, Ars Combin","cited_arxiv_id":null,"evidence_quote":"Gives the structural fact that equidistant linear codes are sequences of dual Hamming codes, used to handle lengths divisible by [k]_q."},{"cited_title":"McKay and A","cited_arxiv_id":null,"evidence_quote":"Supplies graph-isomorphism testing used to decide equivalence among the codes found in the exhaustive searches."},{"cited_title":"Shoup, NTL: A Library for doing Number Theory, Available onlin e at http://www.shoup.net/ntl/","cited_arxiv_id":null,"evidence_quote":"Supplies the determinant computation routine used to check nonsingularity of G G^T in the search program."}],"review_version":1}