REVIEW 5 major objections 5 minor 17 references
On the minimum weights of binary LCD codes and ternary LCD codes
T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper gives exact largest minimum weights for many infinite families of binary and ternary LCD codes.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (5)
- [Section 4.2 (after Table 2, leading to Theorem 4.4)] 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 5 (after Table 5, leading to Theorem 5.5)] 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 7 (Propositions 7.2, 7.5, 7.6)] 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 2.2 (Proposition 2.5 and Lemma 2.2)] 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.
- [Remark 2.4 and the enumeration paragraph in Section 4.2 (and analogous text in Section 5)] 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.
minor comments (5)
- [Section 4.2, paragraph listing r values] 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 6, Table 7] 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.
- [Proof of Proposition 3.4] 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.
- [Proof of Lemma 7.3] 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 1, final paragraph] 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.
Circularity Check
No significant circularity; the paper's results follow from explicit code constructions, Griesmer-bound obstructions, and finite exhaustive searches, with self-citations used as external theorems rather than as fitted inputs.
full rationale
I find no circular derivation in this paper. The main lower bounds are supported by explicitly listed generator matrices or multiplicity vectors (e.g., Tables 4 and 6, Figure 1, and the matrices M26 and M36), which are concrete objects that can be checked independently. The upper bounds come from the Griesmer bound, Ward's divisibility theorem as used in Lemma 3.1, and finite nonexistence assertions from exhaustive computer searches; none of these is fitted to the claimed values. Proposition 2.5, imported from the authors' companion paper [3], is load-bearing for converting finite checks into infinite-family statements, but it is a parameter-free theorem with stated assumptions that do not include the target results, so under the review rules it counts as independent evidence rather than circularity. The same applies to the other self-citations ([1], [2], [9], [10], [11]): they provide prior constructions and classifications, not conclusions equivalent to this paper's theorems. The serious caveat is verification: the exhaustive searches are asserted without shipped programs, logs, or certificates, so a pruning or enumeration bug could change the stated values; however, that is a correctness and reproducibility risk, not circularity. Accordingly, the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (7)
- standard math Griesmer bound for linear codes
- standard math Massey's characterization: code is LCD iff GGT is nonsingular
- domain assumption Lemma 2.2 (LCD construction by prepending simplex blocks) from [3]
- domain assumption Proposition 2.5 (nonexistence reduction) from [3]
- standard math Ward's divisibility of codes meeting the Griesmer bound [17]
- domain assumption Known base values d2/d3 from prior literature [1,2,10,11,16]
- ad hoc to paper Completeness and correctness of the authors' exhaustive computer searches
Cite this review
Pith. "Pith review of On the minimum weights of binary LCD codes and ternary LCD codes." pith.science (2026). https://pith.science/paper/IXEAZBIE
@misc{pith2026190808661,
author = {Pith},
title = {Pith review of: On the minimum weights of binary LCD codes and ternary LCD codes},
year = {2026},
howpublished = {\url{https://pith.science/paper/IXEAZBIE}},
note = {Machine review of arXiv:1908.08661}
}
abstract
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
Figures
Reference graph
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