REVIEW 2 major objections 3 minor 7 references
Remark on subcodes of linear complementary dual codes
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Every ternary Euclidean LCD code contains a Euclidean LCD subcode of dimension one less, and the same holds for quaternary Hermitian LCD codes.
desk verdict A correct, short structural result for ternary and quaternary LCD codes; the stress-test counterexample against the quaternary half is itself wrong, so the paper deserves a normal refereeing pass with minor requests for clarification. 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 argument rests on two standard characterizations. Lemma 2 says that a code is LCD if and only if, for a generator matrix $G$, the matrix $GG^T$ (in the Euclidean case) or $G\overline{G}^T$ (in the Hermitian case) is nonsingular. Lemma 3 says that a ternary code is Euclidean self-orthogonal exactly when every codeword weight is a multiple of three, and a quaternary code is Hermitian self-orthogonal exactly when every codeword weight is even. These two facts combine to give the construction: an LCD code is not self-orthogonal, so by Lemma 3 there is a codeword $x$ whose weight is not divisible by the relevant prime, which implies $\langle x,x\rangle \ne 0$. The proof then modifies the remaining generator rows by adding multiples of a row with $\langle x,z_k\rangle=1$ to force orthogonality with $x$, yielding a generator matrix whose Gram matrix is block diagonal with $\langle x,x\rangle$ and $G_0G_0^*$. Nonsingularity of the whole Gram matrix forces $G_0G_0^*$ nonsingular, so Lemma 2 certifies the subcode generated by $G_0$ as LCD. This block-diagonalization of the Gram matrix is the mechanism that carries the argument.
What would settle it
Search exhaustively over all ternary Euclidean LCD codes up to equivalence for small lengths $n$, say $n \le 10$, and every $k$ with $2 \le k \le n$, checking whether each has a Euclidean LCD $[n,k-1]$ subcode; any code lacking such a subcode would disprove Proposition 4(i). The analogous check for quaternary Hermitian LCD codes tests Proposition 4(ii), and a single counterexample at small $n$ would disprove the theorem.
Extended reading notes
Core claim
The central structural claim is Proposition 4: any ternary Euclidean LCD $[n,k]$ code contains a Euclidean LCD $[n,k-1]$ subcode, and any quaternary Hermitian LCD $[n,k]$ code contains a Hermitian LCD $[n,k-1]$ subcode, for $2 \le k \le n$. Theorem 1 then follows immediately by taking subcodes with larger minimum weight. The proof uses Lemma 3, which characterizes self-orthogonal ternary codes as those whose nonzero codewords all have weight divisible by three, and self-orthogonal quaternary codes as those whose nonzero codewords all have even weight. Since an LCD code is not self-orthogonal, this guarantees a codeword $x$ with $\langle x,x\rangle \ne 0$. After arranging a generator matrix with $x$ as the first row and all other rows orthogonal to $x$, the Gram matrix becomes block diagonal with blocks $\langle x,x\rangle$ and $G_0G_0^*$. Since the full Gram matrix is nonsingular and $\langle x,x\rangle \ne 0$, the subcode generated by the remaining rows is also LCD. The immediate numerical consequence is the monotonicity $d^{E}_3(n,k) \le d^{E}_3(n,k-1)$ and $d^{H}_4(n,k) \le d^{H}_4(n,k-1)$. Moreover, using the fact that every linear code over $\mathbb{F}_q$ for $q \ge 4$ (Euclidean) and over $\mathbb{F}_{q^2}$ for $q \ge 3$ (Hermitian) is equivalent to an LCD code, the same monotonicity is extended to all those fields as Corollary 7.
Load-bearing premise
The load-bearing premise is that Lemma 3 is correct: in ternary codes Euclidean self-orthogonality coincides with all codeword weights being multiples of three, and in quaternary codes Hermitian self-orthogonality coincides with all codeword weights being even; if either characterization failed, the proof could not find the codeword $x$ with $\langle x,x\rangle \ne 0$ that the construction needs.
Editorial extensions
If this is right
- For ternary Euclidean and quaternary Hermitian LCD codes, the optimal minimum distance table is monotone: $d^{E}_3(n,k) \le d^{E}_3(n,k-1)$ and $d^{H}_4(n,k) \le d^{H}_4(n,k-1)$ for every $2 \le k \le n$.
- By code equivalence, the same monotonicity holds for Euclidean LCD codes over every field of order $q \ge 4$ and for Hermitian LCD codes over every field of order $q^2$ with $q \ge 3$.
- Every ternary Euclidean or quaternary Hermitian LCD $[n,k]$ code with $1 \le k \le n-1$ can be extended to an LCD $[n,k+1]$ code containing it, so the family is also closed upward in dimension.
- The proof is constructive: from any generator matrix of an LCD code, it explicitly writes down a generator matrix of an LCD subcode of dimension $k-1$.
Reading between the lines
- The same block-diagonalization strategy may adapt to binary Euclidean LCD codes, since $\langle x,x\rangle = \mathrm{wt}(x) \bmod 2$; the known binary results were established separately for odd and even $k$, and a unified proof could simplify the theory.
- The downward and upward closure together imply that, for these alphabets, there exists a chain of nested LCD codes with every dimension from $1$ to $n$, which may be useful for constructing families of codes with prescribed parameters.
- The monotonicity of $d(n,k)$ means that to determine the full optimal-distance table for a fixed length $n$, one can compute the value at a single dimension and bound the rest by the inequality, reducing the computational search space.
- The proof's hinge is the weight-divisibility characterization, so any alphabet or inner product that admits a similar characterization would immediately inherit the same subcode and monotonicity results.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that for 2 ≤ k ≤ n, every ternary Euclidean LCD [n,k] code contains a Euclidean LCD [n,k−1] subcode, and every quaternary Hermitian LCD [n,k] code contains a Hermitian LCD [n,k−1] subcode. The proof selects a codeword x of nonzero norm using the Huffman–Pless weight characterization of self-orthogonal codes, then uses elementary row operations on a generator matrix to make all other rows orthogonal to x. The resulting block-diagonal Gram matrix shows that the remaining rows generate an LCD subcode of dimension k−1. From this, the monotonicity of the largest minimum weights dE_3(n,k) and dH_4(n,k) follows immediately, and a dual argument yields LCD [n,k+1] supercodes. A final corollary extends the monotonicity to q>3 and q2>4 using the Carlet–Mesnager–Tang–Qi–Pellikaan equivalence result.
Significance. If correct, the paper gives a short and clean proof of the monotonicity of largest minimum weights for ternary Euclidean and quaternary Hermitian LCD codes, complementing the previously known binary case. The argument is constructive and self-contained modulo two standard results: Massey's generator-matrix characterization and the Huffman–Pless weight characterization. The key insight is that in the ternary Euclidean and quaternary Hermitian settings, the weight characterization is strong enough to produce a codeword with nonzero norm, which is exactly the mechanism that the binary case lacks. I also checked the delicate quaternary part: Lemma 3(ii) is correct. The apparent counterexample C = span{(1,1),(ω,ω^2)} over F4 is not a counterexample, because ω(1,1)+(ω,ω^2)=(0,1), so C contains a weight-1 codeword and does not have all weights even.
major comments (2)
- [§3 (proof of Proposition 4)] The orthogonalization step is the load-bearing point of the proof, but the equality ⟨x, z_k+(p−1)⟨x,x⟩x⟩=0 is asserted without justification. It is true, but the authors should supply the one-line verification: over F3 every nonzero square is 1, and over F4 the nonzero value ⟨x,x⟩ lies in F2 and is its own square, so ⟨x,z_k⟩+(p−1)⟨x,x⟩^2 = 1+(p−1) = 0 for p=3 and p=2. The same observation underlies the orthogonality of the other modified rows. As written, the central construction leaves this step to the reader.
- [§3 (generator-matrix reduction to G1–G3)] The reduction to the three forms G1–G3 is correct but is summarized rather than demonstrated. The authors should state explicitly that each basis row with nonzero inner product with x is rescaled so that the inner product becomes 1, that rows with zero inner product form the y-block, and that the ordering is just a permutation. It would also help to note that the transformed non-x rows have rank k−1: the row operations are elementary, and x cannot lie in their span because ⟨x,x⟩ ≠ 0.
minor comments (3)
- [Abstract] There is a typographical spacing error in 'Hermit ian' in the abstract; please fix it.
- [§2, Lemma 2 and Table 1] The notation for the Hermitian adjoint is hard to read in the current typesetting: the second 'GG^T' in Lemma 2 should be G times the conjugate transpose of G. Please define G^* = \bar{G}^T explicitly for the Hermitian case and use it consistently.
- [§4, proof of Proposition 5] When k = n−1, the proof invokes Proposition 4 with dimension n−k−1 = 0, which is outside the stated range 2 ≤ k ≤ n of Proposition 4. The zero code is trivially LCD, so the argument can be repaired with one sentence; please add that remark.
Circularity Check
No circularity: the ternary and quaternary subcode claims are derived from external standard characterizations (Massey, Huffman–Pless) with no self-referential reductions.
full rationale
The paper's derivation chain is: define LCD duality (Section 2), cite Massey's generator-matrix characterization (Lemma 2), cite Huffman–Pless Theorem 1.4.10 (Lemma 3) to obtain a codeword x with weight not divisible by p and hence nonzero self-inner-product, perform an explicit row-reduction/triangulation of the generator matrix, and then apply Massey's criterion to the resulting (k−1)-row submatrix. The conclusion dE_3(n,k) ≤ dE_3(n,k−1) and dH_4(n,k) ≤ dH_4(n,k−1) follows directly from the existence statement in Proposition 4. No parameter is fitted to the target bounds; no quantity in Theorem 1 is used to define Lemma 2 or Lemma 3; and the self-citations (to [1]–[3] for context on binary LCD bounds and equivalence results) are not load-bearing for the main proof. The only external load-bearing input, Lemma 3, is quoted from the textbook [4] and is independent of the paper's own results, so even if the quaternary half of Lemma 3 is mathematically incorrect—as an explicit F4 counterexample suggests—that is a correctness defect, not a circularity defect. The proof does not define 'LCD subcode' in terms of the bound, nor does it rename a known result; it gives a constructive subcode generation argument. Therefore the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Massey's Lemma 2: A code C over F_q (or F_q^2) is LCD iff GG* is nonsingular for a generator matrix G.
- standard math Lemma 3 from Huffman-Pless [4, Thm 1.4.10]: ternary Euclidean self-orthogonal codes are exactly those with all weights divisible by 3, and quaternary Hermitian self-orthogonal codes are exactly those with all weights even.
- domain assumption For nonzero ⟨x,x⟩ in F_3 (Euclidean) or F_4 (Hermitian), ⟨x,x⟩^2 = 1.
- domain assumption Equivalence result of Carlet et al. [2]: every code over F_q is equivalent to a Euclidean LCD code for q ≥ 4, and every code over F_q^2 is equivalent to a Hermitian LCD code for q ≥ 3.
Cite this review
Pith. "Pith review of Remark on subcodes of linear complementary dual codes." pith.science (2026). https://pith.science/paper/XCXTXPQ7
@misc{pith2026190808662,
author = {Pith},
title = {Pith review of: Remark on subcodes of linear complementary dual codes},
year = {2026},
howpublished = {\url{https://pith.science/paper/XCXTXPQ7}},
note = {Machine review of arXiv:1908.08662}
}
abstract
We show that any ternary Euclidean (resp.\ quaternary Hermitian) linear complementary dual $[n,k]$ code contains a Euclidean (resp.\ Hermitian) linear complementary dual $[n,k-1]$ subcode for $2 \le k \le n$. As a consequence, we derive a bound on the largest minimum weights among ternary Euclidean linear complementary dual codes and quaternary Hermitian linear complementary dual codes.
Reference graph
Works this paper leans on
- [1]
- [2]
-
[3]
L. Galvez, J.-L. Kim, N. Lee, Y.G. Roe and B.-S. Won, Some bounds on binary LCD codes, Cryptogr.\ Commun. 10 (2018), 719--728
work page 2018
-
[4]
W.C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003
work page 2003
- [5]
- [6]
-
[7]
Massey, Linear codes with complementary duals, Discrete Math
J.L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992), 337--342
work page 1992
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.