{"id":"491e5345-0177-4929-980a-27741f192eed","arxiv_id":"1908.08661","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact values and bounds for the optimal minimum weight of binary and ternary linear complementary dual codes are established for new infinite families, and several previously published values are corrected.","lead":"This paper determines the largest possible minimum distance of certain binary and ternary 'LCD' error-correcting codes for several infinite families of lengths and dimensions, and it fixes errors in an earlier published table. The exact values are found by combining the Griesmer bound, a code-construction lemma, a nonexistence lemma from divisibility theory, and exhaustive computer searches.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 4.4, 5.5, and 7.2 rest on unverified exhaustive searches; without search artifacts, a pruning bug or missed equivalence class would shift the claimed d2(n,5) or d3(n,4) values.","rationale":"I read the full paper and found the analytical structure coherent: the Griesmer bound, Lemma 3.1, the explicit constructions, and the corrections to [2] are all plausible, and the lower bounds are backed by concrete generator matrices or previously tabulated codes. The main point of vulnerability is exactly the one the reader identified: the infinite-family upper bounds are inherited from finite nonexistence checks that are asserted but not shipped. The paper itself calls these searches exhaustive and gives runtimes, but no code or certificate is made available, and the WLOG reduction in Remark 2.4 is not proved in this paper. A failure in either the search implementation or the reduction would change specific entries in Theorems 4.4 and 5.5, not just weaken bounds. I also note the reliance on Proposition 2.5 from the companion paper; that is a second support gap, but the finite searches are the more immediately checkable and least documented link. My assessment does not move the reader's verdict: CONDITIONAL is the right call pending verifiable artifacts and public availability of the companion paper.","tokens_in":18771,"tokens_out":11973,"duration_ms":119338,"concrete_test":"Independently re-run the enumeration for the binary [2r,5,r] cases with r∈{16,23,24,27,28,29,30,31} and for the ternary [3r,4,2r] cases with r∈{13,22,25,26}, using a separately written backtracker over the vector m in (1) with the constraints of Remark 2.4. Start with r=16 and r=13 as tractable sanity checks; for the expensive cases, an exact-cover or SAT formulation of the same m-vector search would provide an independent yes/no answer. If any code is found where the paper asserts nonexistence, the corresponding residue-class value in Theorem 4.4 or 5.5 must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central upper bounds are not derived purely analytically. In Section 4.2 the paper asserts that \"our exhaustive computer search shows that there is no binary LCD [2r,5,r] code with dual distance d⊥>=2 for only r in {16,23,24,27,28,29,30,31}\", and Section 5 makes the analogous ternary assertion for r in {13,22,25,26}; Section 7.1 adds the nonexistence of binary [n,n−5,3] codes for n=27,...,31. These assertions are the load-bearing premises for Theorem 4.4, Theorem 5.5, and Proposition 7.2. No search program, log, certificate, or independent verifier is provided, and the reported runtimes (e.g., about 1999 core-days for r=24) make casual rechecking impractical. The searches also rely on the WLOG reduction in Remark 2.4 (mi≥1 for selected basis columns and first-row weight d); if that reduction omits an equivalence class, the nonexistence conclusions are unsound. A single false negative among the listed r values would change the corresponding residue-class formula by 1 or 2. The companion Proposition 2.5 is the engine that turns these finite checks into infinite-family statements, so its correctness is also load-bearing, but the finite searches are the least externally verified link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the largest minimum weights d2(n,k) and d3(n,k) of binary and ternary linear complementary dual (LCD) codes. For dimension 5 binary codes and dimension 4 ternary codes, it determines d2(n,5) for 13 residue classes modulo 31 and d3(n,4) for 22 residue classes modulo 40, with bounds for the remaining classes. It also completely determines d2(n,n−5), d3(n,n−2), d3(n,n−3), and d3(n,n−4) for arbitrary n, and d3(n,k) for n=11,...,19. The methods combine the Griesmer bound with a nonexistence lemma for LCD codes meeting it, a construction that extends codes by simplex-code blocks, a reduction proposition (Proposition 2.5) that converts finite nonexistence checks into infinite-family statements, and several exhaustive computer searches.","tokens_in":18979,"tokens_out":10082,"duration_ms":94893,"significance":"If correct, the results are a substantial contribution: they settle optimal minimum distances for infinite families of binary and ternary LCD codes and complete the picture for high-rate binary and ternary LCD codes in several dimensions. The analytical parts are transparent: Lemma 3.1 and its applications in Propositions 3.3 and 3.4 are clean, and the explicit generator matrices and vectors in Tables 4, 6, and Figure 1 provide checkable constructions for the lower bounds. The paper also corrects an error in the authors' earlier work, which is a useful service. However, the decisive upper bounds in Theorems 4.4 and 5.5, and the high-rate determinations in Section 7, rest on exhaustive computer searches for which no program, log, or certificate is provided, and on Proposition 2.5 from an unpublished companion paper. These load-bearing elements must be made auditable before the results can be fully accepted.","major_comments":[{"comment":"The upper bounds for d2(n,5) depend on the asserted exhaustive search that there is no binary LCD [2r,5,r] code with dual distance d⊥≥2 exactly for r ∈ {16,23,24,27,28,29,30,31}, with about 1999 core-days reported for r=24. No search program, log, or certificate is shipped, and no independent verifier is cited. A single false negative among these r values would shift the corresponding residue-class formula in Theorem 4.4 by 1 or 2, so this is a load-bearing premise. Please supply the search code, logs, or a machine-checkable certificate, or an independently reproduced verification.","section":"Section 4.2 (after Table 2, leading to Theorem 4.4)"},{"comment":"The same issue arises for the ternary assertion that there is no ternary LCD [3r,4,2r] code with dual distance d⊥≥2 exactly for r ∈ {13,22,25,26}, with about 709 core-days reported for r=22. Proposition 5.4 and Theorem 5.5 depend on this assertion. As with the binary case, the absence of search artifacts makes the central upper bounds currently unverifiable by the reader or referee.","section":"Section 5 (after Table 5, leading to Theorem 5.5)"},{"comment":"The complete determinations d2(n,n−5)=2 for n≥27, d3(n,n−3)=2 for n≥11, and d3(n,n−4)=2 for n≥37 rely on the assertions that no binary [n,n−5,3] code exists for n=27,...,31, and that no ternary [n,n−3,3] code is LCD for n=11,12,13 and no ternary [n,n−4,3] code is LCD for n=37,38,39. These exhaustive searches are described only in words, with no code, log, or certificate. Because these claims separate the value 2 from larger values in infinite families, they need to be auditable.","section":"Section 7 (Propositions 7.2, 7.5, 7.6)"},{"comment":"Proposition 2.5, which converts the finite nonexistence checks into infinite-family upper bounds in Theorems 4.4 and 5.5, is quoted from the companion paper [3] (arXiv:1908.03294, submitted). Lemma 2.2, used for all lower-bound constructions, comes from the same companion. If [3] is not yet published, please include full proofs of these results in the manuscript or a clearly marked appendix, so that the main theorems do not rest on an unavailable reference.","section":"Section 2.2 (Proposition 2.5 and Lemma 2.2)"},{"comment":"The claim that one may assume without loss of generality that m_i ≥ 1 for selected basis columns and that ∑_{i∈S} m_i = d is asserted without proof. Since the exhaustive searches enumerate only these representatives, an omitted equivalence class would invalidate the nonexistence conclusions. Please provide a proof or a precise citation establishing this reduction.","section":"Remark 2.4 and the enumeration paragraph in Section 4.2 (and analogous text in Section 5)"}],"minor_comments":[{"comment":"The phrase 'for only r ∈ {...}' is ambiguous; it should be rephrased as 'only for r ∈ {...}' or 'for exactly the r-values ...' to avoid the impression that the search was run for other r and found codes.","section":"Section 4.2, paragraph listing r values"},{"comment":"Entries with two values, such as '8,9' for n=20, k=8, are not explained in the text. Please state explicitly which entries are the four exceptions and whether these entries are unresolved possible values or final values with a range.","section":"Section 6, Table 7"},{"comment":"The proof begins with 'Let s be a positive integer' while the statement is for s nonnegative. Please cover s=0 explicitly, or state that the s=0 values are taken from the cited tables.","section":"Proof of Proposition 3.4"},{"comment":"The sentence 'By the construction, it is trivial that C has minimum weight 2' deserves a short justification, especially for the i=2 case where the displayed parity-check matrix has repeated columns.","section":"Proof of Lemma 7.3"},{"comment":"The paper says all computer calculations were done in C and Magma but gives no indication of where the programs or logs can be obtained. Please provide URLs or include the programs as ancillary material, consistent with the need to audit the exhaustive searches.","section":"Section 1, final paragraph"}],"recommendation":"major_revision","confidential_remarks":"The reported core-day figures suggest the authors did run the searches, but the absence of code, logs, or certificates is a serious reproducibility gap for a paper whose main upper bounds are finite-search statements. The companion paper [3] is central to the method and should be obtained or included. If the authors can provide auditable search artifacts and proofs of the imported propositions, the paper is likely a solid contribution. I recommend major revision rather than acceptance at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — read this one if you care about LCD code tables. It pins down d2(n,5) for several residue classes mod 31, d3(n,4) for several mod 40, and completes d2(n,n−5), d3(n,n−2..4) for arbitrary n. It also catches and corrects an error in the authors’ own earlier paper. The new exact values are genuinely new, and the Griesmer-bound nonexistence lemmas (Lemma 3.1, from Ward) are transparent. Lower bounds come from explicit generator matrices or Lemma 2.2’s concatenation with simplex codes; those I can check, and the paper gives enough data in tables and figures.\n\nThe soft spot is the load-bearing computational upper bounds. Theorems 4.4 and 5.5 depend on asserted exhaustive searches over equivalence classes of LCD codes with dual distance at least 2, and Propositions 7.2, 7.5, 7.6 depend on similar finite checks. No search program, log, or certificate is provided, and runtimes like 1999 core-days mean nobody will casually rerun them. The reduction in Remark 2.4 is plausible but not machine-checked; a pruning bug or missed equivalence class would change the residue-class values by 1 or 2. This is a real weakness under reproducibility standards, though within the conventions of coding theory such computer classifications are often accepted. Proposition 2.5, the engine converting finite checks to infinite families, is imported from a companion paper; self-citation here is not lazy, but it does mean external verification is thin.\n\nProportion: I do not see the central argument collapsing. The analytic parts are tidy, the lower bounds are explicit, and the paper is honest about corrections and exceptions. The unresolved part is verification of the finite searches. I would like to see code or at least machine-readable certificates for the nonexistence claims before trusting the tables fully.\n\nRecommendation: send it to peer review. A serious referee can audit the reductions and request artifacts; desk rejection would lose a useful, correct-looking contribution. Reading group: maybe, if the group is coding theory. I probably will not cite it myself this year, but I would point a student to it for the table and the reduction pattern.","headline":"A solid exact-value contribution to LCD-code tables that deserves refereeing, provided someone asks for the missing search artifacts behind the key upper bounds.","tokens_in":19623,"tokens_out":2321,"would_cite":false,"duration_ms":25307,"reading_group":"maybe","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":"This paper gives exact largest minimum weights for many infinite families of binary and ternary LCD codes.","keywords":["LCD codes","linear complementary dual codes","minimum weight","Griesmer bound","binary codes","ternary codes","optimal codes","simplex codes"],"falsifier":"A reader can try to construct a binary LCD $[31,5,15]$ code: Theorem 4.4 says $d_2(31,5)=14$, so such a code would refute it. For the ternary side, a ternary LCD $[26,4,17]$ code would contradict the claimed $d_3(26,4)=16$. Both parameters are small enough for a direct search or a known-code check.","tokens_in":18487,"feed_emoji":"📏","tokens_out":17110,"duration_ms":150808,"temperature":0.7,"pith_summary":"Linear complementary dual (LCD) codes are error-correcting codes whose only vector common with their dual code is the zero vector; they originated in a multiple-access channel problem and are studied as a class of codes with useful algebraic properties. This paper determines, for many infinite families of lengths, the largest possible minimum weight among binary LCD codes of dimension 5 and ternary LCD codes of dimension 4. It also fixes the optimal minimum weights for every length in three large-codimension families: binary codes of dimension $n-5$ and ternary codes of dimensions $n-2$, $n-3$, and $n-4$. For ternary LCD codes it gives complete tables for lengths 11 through 19 and corrects an earlier published value for one residue class modulo 31. If the reported exhaustive searches are sound, these values close the remaining gaps in those parameter ranges.","feed_headline":"Optimal distances found for infinite families of LCD codes","feed_subtitle":"Exact maximum minimum weights for many binary and ternary code lengths, plus all large-codimension cases.","key_machinery":"The proof is carried by three pieces of machinery. First, the Griesmer bound $n \\ge \\sum_{i=0}^{k-1} \\lceil d/q^i \\rceil$ gives a ceiling $g_q(n,k)$ on the minimum weight, and Lemma 3.1 uses a divisibility theorem to rule out LCD codes that meet it: no binary LCD code of odd dimension and even minimum weight meets the bound, and no ternary LCD code whose minimum weight is divisible by 3 meets it. Second, a stacking construction (Lemma 2.2) appends $s$ copies of the $k$-dimensional simplex code to any LCD code, increasing length by $[k]_q s$ and minimum weight by $q^{k-1}s$, which produces codes attaining the proposed values. Third, Proposition 2.5 converts a finite exhaustive-search statement about small LCD codes with dual distance at least 2 into a nonexistence statement for every $s$, so the reported searches for binary lengths $2r$ and ternary lengths $3r$ are what turn the finite checks into infinite-family upper bounds.","core_discovery":"On the paper's own terms, the central discovery is a collection of exact optimal-distance theorems. Writing binary lengths as $31s+t$, it proves $d_2(n,5)=\\lfloor 16n/31 \\rfloor -1$ for thirteen residue classes $t$ and $d_2(n,5)=\\lfloor 16n/31 \\rfloor -2$ for $t\\in\\{0,6\\}$, with $s\\ge 0$ and $n\\ge 5$. Writing ternary lengths as $40s+t$, it proves $d_3(n,4)=\\lfloor 27n/40 \\rfloor -1$ for twenty residue classes and $d_3(n,4)=\\lfloor 27n/40 \\rfloor -2$ for $t=6$. It completely determines $d_3(n,k)$ for $n\\in\\{11,12,\\ldots,19\\}$, and it determines $d_2(n,n-5)=2$ for $n\\ge 27$, $d_3(n,n-3)=2$ for $n\\ge 11$, and $d_3(n,n-4)=2$ for $n\\ge 37$, with explicit small exceptional values below those thresholds. These results turn previously open intervals into single numbers.","pith_inferences":["A natural extension is to apply the same divisibility obstruction to dimensions 6 and up; the paper already gives isolated exact values for $k=7$ and $k=9$, suggesting a pattern that could be systematized.","The classes left open for $d_2(n,5)$ and $d_3(n,4)$ are open precisely because their finite checks are beyond current exhaustive search; finding a certificate or a theoretical shortcut for the binary $r=24$ case or the ternary $r=22$ case would likely close several residue classes at once.","The small exact table for $d_3(n,k)$ with $11\\le n \\le 19$ is also a constructive resource: any of its entries can serve as a seed for an infinite optimal family, so the table is likely to be reused beyond its stated range.","The correction to an earlier published value hints that other published LCD tables from the same era may contain similar residue-class indexing errors; re-deriving them from column-multiplicity enumeration would be a cheap way to check."],"forward_implications":["The exact values in Theorem 4.4 and Theorem 5.5 can be imported directly into tables of optimal LCD parameters, replacing ranges with single numbers for the covered residue classes.","Because $d_2(n,n-5)$, $d_3(n,n-2)$, $d_3(n,n-3)$, and $d_3(n,n-4)$ are now known for all $n$, classification efforts for large-dimension LCD codes no longer have an open question at these codimensions.","The finite seed codes listed in Section 6, combined with the stacking lemma, yield infinite optimal families such as $d_3(121s+17,5)=81s+9$ and $d_3(364s+13,6)=243s+6$.","The corrected bound for $n\\equiv 12 \\pmod{31}$ changes previously published tables by one unit, and all later values built on that bound inherit the correction."],"supporting_citations":[{"why":"Supplies Lemma 2.2 and Proposition 2.5, which construct high-weight LCD codes and reduce infinite-family upper bounds to finite exhaustive checks.","marker":"[3]"},{"why":"Provides the divisibility theorem from which Lemma 3.1 rules out LCD codes meeting the Griesmer bound.","marker":"[17]"},{"why":"Gives earlier tables and the proposition that is corrected for $n\\equiv 12 \\pmod{31}$.","marker":"[2]"},{"why":"Classifies ternary LCD codes for length at most 10 and supplies base codes used as seeds.","marker":"[1]"},{"why":"Determines $d_2(n,2)$ and contributes low-length tables used as base cases.","marker":"[10]"},{"why":"Determines $d_2(n,3)$ and adds low-length tables used in later sections.","marker":"[11]"},{"why":"Introduces the LCD property and the generator-matrix nonsingularity characterization used throughout.","marker":"[14]"},{"why":"Supplies background on simplex codes and their constant-weight, self-orthogonal properties used in the stacking construction.","marker":"[12]"},{"why":"Shows binary even LCD codes have even dimension, supporting Lemma 3.1(i).","marker":"[7]"},{"why":"Provides known bounds $d^{\\mathrm{all}}_3(n,k)$ used to bound ternary codes in Section 7.","marker":"[5]"}],"fun_headline_variants":["Exact min weights for infinite families of LCD codes","LCD code distances: exact for lengths 11-19 and infinite families","All ternary LCD lengths 11-19 now have exact optimal weights","Floor formulas yield exact min distances for LCD codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole edifice rests on the reliability of the reported exhaustive computer searches, since the proof ships no program, log, or certificate to verify that no small LCD code was missed.","fun_headline_variants_meta":{"raw":{"variants":["Exact min weights for infinite families of LCD codes","LCD code distances: exact for lengths 11-19 and infinite families","All ternary LCD lengths 11-19 now have exact optimal weights","Floor formulas yield exact min distances for LCD codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.003017,"raw_usage":{"total_tokens":11428,"prompt_tokens":933,"completion_tokens":10495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":10426}},"tokens_in":549,"tokens_out":10495,"duration_ms":61740,"temperature":1.0,"reasoning_tokens":10426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:33:23.768461+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A reader can try to construct a binary LCD $[31,5,15]$ code: Theorem 4.4 says $d_2(31,5)=14$, so such a code would refute it. For the ternary side, a ternary LCD $[26,4,17]$ code would contradict the claimed $d_3(26,4)=16$. Both parameters are small enough for a direct search or a known-code check.","supporting_citations":[{"cited_title":"Characterization and classification of optimal LCD codes","cited_arxiv_id":"1908.03294","evidence_quote":"Supplies Lemma 2.2 and Proposition 2.5, which construct high-weight LCD codes and reduce infinite-family upper bounds to finite exhaustive checks."},{"cited_title":"Ward, Divisibility of codes meeting the Griesmer bound, J","cited_arxiv_id":null,"evidence_quote":"Provides the divisibility theorem from which Lemma 3.1 rules out LCD codes meeting the Griesmer bound."},{"cited_title":"Araya and M","cited_arxiv_id":null,"evidence_quote":"Gives earlier tables and the proposition that is corrected for $n\\equiv 12 \\pmod{31}$."},{"cited_title":"Araya and M","cited_arxiv_id":null,"evidence_quote":"Classifies ternary LCD codes for length at most 10 and supplies base codes used as seeds."},{"cited_title":"Galvez, J.-L","cited_arxiv_id":null,"evidence_quote":"Determines $d_2(n,2)$ and contributes low-length tables used as base cases."},{"cited_title":"Harada and K","cited_arxiv_id":null,"evidence_quote":"Determines $d_2(n,3)$ and adds low-length tables used in later sections."},{"cited_title":"Massey, Linear codes with complementary duals, Discrete Math","cited_arxiv_id":null,"evidence_quote":"Introduces the LCD property and the generator-matrix nonsingularity characterization used throughout."},{"cited_title":"Huffman and V","cited_arxiv_id":null,"evidence_quote":"Supplies background on simplex codes and their constant-weight, self-orthogonal properties used in the stacking construction."},{"cited_title":"Carlet, S","cited_arxiv_id":null,"evidence_quote":"Shows binary even LCD codes have even dimension, supporting Lemma 3.1(i)."},{"cited_title":"Brouwer, Bounds on the size of linear codes, Handbook of Co ding Theory, pp","cited_arxiv_id":null,"evidence_quote":"Provides known bounds $d^{\\mathrm{all}}_3(n,k)$ used to bound ternary codes in Section 7."}],"review_version":1}