{"id":"3b6749b8-c3a6-4983-9e63-0de29c2b70d6","arxiv_id":"1908.08662","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any ternary Euclidean and quaternary Hermitian LCD code contains an LCD subcode of one dimension less, implying a bound on maximum minimum weights.","lead":"This paper proves that any ternary Euclidean or quaternary Hermitian linear complementary dual (LCD) code of dimension k contains a smaller LCD subcode of dimension k−1. The result yields monotonicity bounds on the largest minimum weights for these two code families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quaternary half of the main theorem is false: Lemma 3(ii) fails, and an explicit Hermitian LCD [2,2] code has no Hermitian LCD [2,1] subcode.","rationale":"The reader correctly identified Lemma 3 as the load-bearing assumption, but treated it as merely needing a justification. The stress-test shows the lemma is not merely unproven: Lemma 3(ii) is false. The explicit code C = span{(1,1),(ω,ω^2)} over F4 has all nonzero codewords of even weight, but is not Hermitian self-orthogonal, because G G^* = [[0,1],[1,0]] ≠ 0. Since C^⊥ = {0}, C is Hermitian LCD. Every 1-dimensional subcode has generator of weight 2 and therefore has zero Hermitian norm, so no 1-dimensional subcode is LCD. This is a direct counterexample to Proposition 4(ii) at n=k=2, and it also makes the dH_4(2,2) ≤ dH_4(2,1) bound false. Because the quaternary half of the central claim fails, the paper's main theorem cannot stand as stated; the ternary half may still be correct, but the current verdict should be REJECT rather than CONDITIONAL.","tokens_in":4268,"tokens_out":15332,"duration_ms":153581,"concrete_test":"Verify the counterexample explicitly with F4 = {0,1,ω,ω^2}. Let G = [[1,1],[ω,ω^2]] and C be the row space. Compute C^⊥ by solving z1+z2=0 and z1ω^2+z2ω=0, which gives C^⊥={0}. Compute G G^* = [[0,1],[1,0]] ≠ 0. Enumerate the three nonzero 1-dimensional subspaces of C, generated by (1,1), (ω,ω^2), and (ω^2,ω), and compute ⟨v,v⟩ = 1+1 = 0 for each. These computations show C is Hermitian LCD but has no Hermitian LCD [2,1] subcode, refuting Proposition 4(ii) and Theorem 1(ii).","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Proposition 4(ii) depends entirely on Lemma 3(ii), quoted from [4, Theorem 1.4.10]. That lemma is false as stated. Over F4, even Hamming weight does not imply Hermitian self-orthogonality. Let C = span{(1,1),(ω,ω^2)} over F4. Every nonzero codeword is a scalar multiple of (1,1), (ω,ω^2), or (ω^2,ω), so every nonzero codeword has weight 2. For the generator matrix G with rows (1,1) and (ω,ω^2), a direct computation gives G G^* = [[0,1],[1,0]] ≠ 0, so C is not Hermitian self-orthogonal even though all weights are even. Moreover, solving the Hermitian orthogonality equations gives C^⊥ = {0}, so C is Hermitian LCD. But every 1-dimensional subcode of C is generated by a vector of weight 2, whose Hermitian norm is 1+1 = 0, so no 1-dimensional subcode is LCD. Thus C contains no LCD [2,1] subcode, directly contradicting Proposition 4(ii). Consequently dH_4(2,2) ≥ 2 while dH_4(2,1) = 1, so the inequality dH_4(2,2) ≤ dH_4(2,1) in Theorem 1 is false. The paper's step 'By Lemma 3, there is a codeword x with wt(x) not congruent to 0 mod p' has no valid basis in the quaternary Hermitian case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":4626,"tokens_out":19729,"duration_ms":196302,"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":[{"comment":"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.","section":"§3 (proof of Proposition 4)"},{"comment":"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.","section":"§3 (generator-matrix reduction to G1–G3)"}],"minor_comments":[{"comment":"There is a typographical spacing error in 'Hermit ian' in the abstract; please fix it.","section":"Abstract"},{"comment":"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.","section":"§2, Lemma 2 and Table 1"},{"comment":"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.","section":"§4, proof of Proposition 5"}],"recommendation":"minor_revision","confidential_remarks":"The contested point in the review process — the validity of Lemma 3(ii) — is in fact correct; I verified the polarization argument over F4 and found no counterexample. The paper is a short, incremental remark, but the proof is sound and the exposition is clear apart from the local gaps noted in the report. It fits the scope of a coding-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a short note with a clean result: 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 monotonicity bounds on dE_3 and dH_4 follow immediately. The binary analogue was known, so the novelty is a modest but genuine extension to the two hardest remaining small-field cases.\n\nThe proof is sound. The generator-matrix reduction is standard, and the key algebraic step—showing that the modified last row is orthogonal to x—works because for any nonzero norm value over F3 or F4, ⟨x,x⟩^2 = 1. The paper leaves that justification implicit; a referee should ask for the one-line proof. The q>3 corollary also leans on Carlet et al.'s equivalence result rather than on Theorem 1 itself; the wording could be clearer, but the logic is fine.\n\nThe stress-test note's counterexample does not hold up. The code C = span{(1,1),(ω,ω^2)} over F4 is not a counterexample to Lemma 3(ii): it contains (0,1), a weight-1 codeword, so the premise \"all weights even\" fails. Moreover, Lemma 3(ii) itself is true. If every codeword has even weight, then every codeword has zero Hermitian norm; because C is closed under multiplication by ω, polarization forces all pairwise inner products to vanish. The paper's reliance on the Huffman–Pless characterization is legitimate.\n\nThe only real soft spots are cosmetic: the unstated norm-square identity and the slightly loose derivation of Corollary 7. Neither threatens the main theorem. The paper is well written and properly cites the earlier binary work and the relevant characterizations.\n\nWho is this for? Anyone working with LCD code tables or bounds over F3 and F4 will use this monotonicity fact. It is not a landmark, but it is a solid, verifiable building block. A serious referee should engage with it; I would expect a quick conditional acceptance after minor revision.","headline":"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.","tokens_in":5131,"tokens_out":8739,"would_cite":true,"duration_ms":86675,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B65","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every ternary Euclidean LCD code contains a Euclidean LCD subcode of dimension one less, and the same holds for quaternary Hermitian LCD codes.","keywords":["linear complementary dual codes","LCD codes","ternary Euclidean codes","quaternary Hermitian codes","subcodes","largest minimum weight","self-orthogonal codes","code equivalence"],"falsifier":"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.","tokens_in":4081,"feed_emoji":"🧮","tokens_out":14861,"duration_ms":110963,"temperature":0.7,"pith_summary":"This paper proves a structural property of linear complementary dual (LCD) codes over two specific alphabets: over the field with three elements (ternary codes with the Euclidean dot product) and over the field with four elements (quaternary codes with the Hermitian dot product), every LCD code of dimension $k$ contains an LCD subcode of dimension $k-1$, for every $2 \\le k \\le n$. The proof is linear-algebraic and short: it finds a codeword whose norm is nonzero, rewrites the generator matrix so that all other rows are orthogonal to that codeword, and shows the remaining rows still generate an LCD code. Because subcodes have minimum weight at least that of the parent code, the largest achievable minimum weights satisfy $d^{E}_3(n,k) \\le d^{E}_3(n,k-1)$ and $d^{H}_4(n,k) \\le d^{H}_4(n,k-1)$. The same argument also shows every such LCD code is contained in an LCD code of dimension one larger, so the family of LCD codes is closed under both shrinking and growing by one dimension.","feed_headline":"Ternary and quaternary LCD codes always have LCD subcodes","feed_subtitle":"For these alphabets, the best achievable minimum weight cannot increase as the code dimension grows.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $k-1$."],"supporting_citations":[{"why":"It supplies the generator-matrix criterion for a code to be LCD, which is used to certify that the constructed subcode is LCD.","marker":"[5]"},{"why":"It supplies the weight-divisibility characterization of self-orthogonal ternary and quaternary codes, which provides the codeword with nonzero norm.","marker":"[4, Theorem 1.4.10]"},{"why":"It shows that every linear code over $\\mathbb{F}_q$ with $q \\ge 4$ (or over $\\mathbb{F}_{q^2}$ with $q \\ge 3$) is equivalent to an LCD code, which extends the monotonicity bound to those fields.","marker":"[2]"}],"fun_headline_variants":["Every ternary and quaternary LCD code has an LCD subcode","LCD codes over GF(3) and GF(4) always contain LCD subcodes","For ternary and quaternary LCD codes, an LCD subcode always exists","Monotonicity of best LCD weights follows from subcode existence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every ternary and quaternary LCD code has an LCD subcode","LCD codes over GF(3) and GF(4) always contain LCD subcodes","For ternary and quaternary LCD codes, an LCD subcode always exists","Monotonicity of best LCD weights follows from subcode existence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001475,"raw_usage":{"total_tokens":5933,"prompt_tokens":957,"completion_tokens":4976,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":4897}},"tokens_in":573,"tokens_out":4976,"duration_ms":38373,"temperature":1.0,"reasoning_tokens":4897,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:32:42.307631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Kennedy and V","cited_arxiv_id":null,"evidence_quote":"It supplies the generator-matrix criterion for a code to be LCD, which is used to certify that the constructed subcode is LCD."},{"cited_title":"Carlet, S","cited_arxiv_id":null,"evidence_quote":"It shows that every linear code over $\\mathbb{F}_q$ with $q \\ge 4$ (or over $\\mathbb{F}_{q^2}$ with $q \\ge 3$) is equivalent to an LCD code, which extends the monotonicity bound to those fields."}],"review_version":1}