{"id":"3fcd8e54-3c19-4433-a26a-fd6586eff099","arxiv_id":"2507.07842","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New lower bounds for constant dimension codes A_q(n,2δ,{k}) are obtained by adding a generalized bilateral multilevel construction onto the parallel mixed dimension construction.","lead":"This paper gives a recipe for building larger constant dimension codes, a type of error-correcting code used in random network coding. The authors combine two recent construction methods and report dozens of improved size records for such codes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's proof is omitted and the identity behind Theorem 4's Case 2 is unproved; the identity is likely valid, but the Theorem 7 lower bounds remain conditional until the proof is supplied.","rationale":"I checked the central construction and its arithmetic. Example 6 and Example 7 both recompute correctly from the stated formulas, and the new bounds in Table 2 are consistent with Corollaries 2 and 3. The specific identity flagged by the reader in Theorem 4's Case 2 is true: in reduced row inverse echelon form, rows have zeros to the right of their pivots, so the number of pivots in a suffix of columns equals the dimension of the projection onto that suffix; and the block structure of the generalized bilateral echelon form makes rank(V2) = wt(bar v2) + rank(phi_{\\tilde v}(M)). Thus the reader's weakest assumption is not a correctness flaw, but it is an unproved assertion in the manuscript. The omitted proof of Theorem 5 is a more substantial gap because Theorem 7 and its corollaries depend on it, and the paper provides no argument beyond 'similar to Theorem 4'. The numerical examples do not cover all parameter ranges claimed in Corollaries 2 and 5, so a written proof of Theorem 5 is needed before the new lower bounds for n = 13..17 can be regarded as fully established. I therefore keep the reader's conditional verdict, with the concern narrowed to the missing proof rather than an actual false identity.","tokens_in":20525,"tokens_out":33452,"duration_ms":360869,"concrete_test":"Add a lemma: for any k x n generator in generalized bilateral echelon form, if the last mu_2 columns form V2, then the reduced row inverse echelon form of the row space has exactly rank(V2) pivots in those columns. Prove Theorem 5 by copying the Case 1 and Case 2 argument of Theorem 4 with this lemma. As an independent numerical check, for q=2, n=15, k=4, delta=2 and each M used in Example 7, compute the RRIEF and verify wt(bar v'_2) = rank(V2) = wt(bar v2) + rank(phi(M)); a single violation would invalidate the distance bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 4, Case 2, the proof asserts without justification that the inverse identifying vector of a bilaterally lifted codeword V satisfies wt(bar v'_2) = rank(V2) = wt(bar v2) + rank(phi_{\\tilde v}(M)), where V2 is the last mu_2 columns. This identity is load-bearing: it yields dS(U,V) >= 2 delta for U in C3. The first equality is a general reduced-row-inverse-echelon fact (rows have zeros to the right of their pivot, so the number of pivots in a suffix equals the rank of the projection onto that suffix), and the second follows from the zero pattern in the pivot columns of EF(bar v2); so the assertion is correct, but it is not demonstrated. More importantly, Theorem 5, which underlies Theorem 7 and hence Corollaries 2 and 5 and five of the seven Table 2 entries, has its proof omitted with only 'similar to Theorem 4'. Because Theorem 7 uses a different bilateral vector pattern (with mu_2 = k and omega_2 = delta) than Theorem 6, the analogous distance check is not literally identical, and the new lower bounds for A2(13..17) are not fully established as written. This warrants a conditional verdict pending a written proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper combines the generalized bilateral multilevel construction of Hong--Cao--Luo with the parallel mixed dimension construction of He et al. to produce constant-dimension subspace codes. The main new ingredient is a set of bilateral identifying vectors whose lifted generalized bilateral Ferrers diagram rank-metric codes can be added to the union C1 ∪ C3 of the parallel mixed dimension construction without violating the minimum distance 2δ. Theorems 4 and 6 give conditions and explicit size formulas for the enlarged code, and Theorems 5 and 7 give a simplified version with n3 = k and T2 = {k}. The authors report improved lower bounds, for example A2(18,4,{4}) ≥ 5199103860464, and state that at least 49 new lower bounds are obtained. The size formulas are derived explicitly from prior theorems, with no fitted parameters.","tokens_in":20783,"tokens_out":34563,"duration_ms":314384,"significance":"If the proofs are completed, the paper gives a systematic and apparently effective way to improve the parallel mixed dimension construction, with many concrete numerical improvements over the best-known bounds in [16]. The construction is transparent: the added bilaterally lifted code is disjoint from C1 ∪ C3 by construction, and its size is computed from explicit rank-metric formulas, so there is no circularity or parameter fitting. The numerical examples in Examples 6--8 are internally consistent with the displayed formulas. The significance is conditional on two proof gaps: an asserted but unproved rank identity in the proof of Theorem 4, and the entirely omitted proof of Theorem 5, which underlies Theorem 7 and several table entries.","major_comments":[{"comment":"The proof of Theorem 5 is omitted with the sentence 'The proof is similar to that of Theorem 4 and is omitted here.' This is not a merely cosmetic omission: Theorem 5 is the basis for Theorem 7, Corollaries 2 and 5, and therefore for five of the seven q = 2 entries and the q = 3 entries with 13 ≤ n ≤ 17 in Table 2. The parameter pattern in Theorem 7 differs from that in Theorem 6 (μ2 = k and ω2 = δ instead of general μ2 and ω2), so the distance verification for C3′ ∪ tilde-C1′ is not literally identical to the proof of Theorem 4. The authors should provide the full proof of Theorem 5, in particular the Case 2 computation showing that dS(U,V) ≥ 2δ for U ∈ C3′ and V ∈ tilde-C1′.","section":"§3, Theorem 5"},{"comment":"In Case 2 of the proof of Theorem 4, the paper asserts without proof that wt(bar-v2′) = rank(V2) = wt(bar-v2) + rank(phi-tilde-v(M)), where V2 is the last μ2 columns of the generator matrix V obtained by lifting a matrix M from the GB-FD code. This identity is load-bearing: it is exactly what converts the inverse-identifying-vector distance into 2·|wt(bar-v2′) − wt(bar-u2)| and hence yields dS(U,V) ≥ 2δ for U ∈ C3. The statement is plausible and can be proved using the block structure of gEF(tilde-v) together with the fact that the number of inverse-RREF pivots in a coordinate suffix equals the rank of the projection onto that suffix, but the proof should be written out. As it stands, the minimum-distance guarantee for C3 ∪ tilde-C1 is not fully established.","section":"§3, proof of Theorem 4, Case 2"}],"minor_comments":[{"comment":"The displayed matrix bar-E(U) does not satisfy the definition of reduced row inverse echelon form: the first and second rows both have their leading 1 in column 1, which violates the requirement that leading coefficients are the only nonzero entries in their columns and occur in distinct positions. Please correct the example or its typesetting.","section":"Example 2"},{"comment":"In the proof of Theorem 6, the notation 'fEF(tilde-v_{i,j})' should be 'gEF(tilde-v_{i,j})' for consistency with Definition 5.","section":"Proof of Theorem 6"},{"comment":"The assertions 'wt(u1) = k' and 'wt(bar-u2) = k' are stated as clear. They follow from the block form of the parallel mixed dimension construction, but a one-sentence justification should be added, especially because the MRD/RRMC blocks P and Q′ might a priori introduce pivots outside the first μ1 and last μ2 columns.","section":"Proof of Theorem 4, Case 1"},{"comment":"In the proofs of Theorems 6 and 7, the hypotheses of Proposition 1 (in particular l1 ≥ δ and l3 ≥ δ for every i,j in the stated ranges) are not explicitly verified. The choices of θ1 and θ2 do ensure these inequalities, but the verification should be included for completeness.","section":"Application of Proposition 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a natural and mostly transparent application of two recent constructions, and the numerical claims appear consistent with the formulas. The main issue is completeness: the omitted proof of Theorem 5 and the unproved rank identity in Theorem 4 directly support Table 2, so the lower bounds are conditional as written. If the authors supply the missing proofs, I would expect the paper to be acceptable. No circularity or parameter fitting is apparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on the paper. The core idea is legitimate: combine the generalized bilateral multilevel construction with the parallel mixed dimension construction, and choose the bilateral identifying vectors so that the added code is disjoint and at the right distance from the existing union. The authors derive explicit size formulas (Theorems 6 and 7) and produce dozens of new entries for A_q(n,4,{k}) tables, with q up to 9 and n up to 19. The arithmetic in the examples checks out, and the new code is added disjointly, so the improvement is additive and not circular. No free parameters are fitted.\n\nWhat the paper does well: it states the conditions on the identifying vectors precisely, and it gives a full proof of Theorem 4 (modulo one gap noted below) and Theorem 6. The comparison examples are transparent and reproducible.\n\nThe soft spots, in order of importance. First, Theorem 5's proof is omitted with 'similar to Theorem 4'. That's not just a formality: Theorem 7 uses a different bilateral vector pattern (mu2 = k, omega2 = delta), so the analogous distance check is not literally identical. Five of the seven Table 2 entries and Corollaries 2 and 5 rest on Theorem 7. The proof needs to be written out. Second, in Theorem 4, Case 2, the identity wt(bar v'_2) = wt(bar v2) + rank(phi(M)) is asserted without proof. It is load-bearing for the distance from the new code to C3. It is likely true -- the reduced-row-inverse-echelon structure gives the first equality, and the zero pattern in the pivot columns of EF(bar v2) gives the second -- but it is not demonstrated. A referee should ask for one paragraph there. Third, the paper claims 'best-known' but only compares with [16], not with the subspacecodes tables; some of the claimed entries may already be known from other constructions. The numerical gains are also tiny relative to the code sizes (about 2.4 million on a 5.2 trillion bound), so the practical impact is modest even if the bounds stand.\n\nWho is this for? People who track lower bounds for constant dimension codes and want to know the current table entries. It's a niche but real contribution. My recommendation: send it to peer review, but require the omitted proof of Theorem 5 and the justification of the identity in Theorem 4. With those supplied, the paper is solid.","headline":"A sound, incremental lower-bound construction for CDCs; proof gaps in Theorems 4 and 5 should be fixed before the results are treated as established.","tokens_in":21313,"tokens_out":3436,"would_cite":true,"duration_ms":33994,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B60","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims a bilateral lifting construction that enlarges many best-known constant dimension codes by joining a generalized bilateral multilevel code to the parallel mixed dimension construction.","keywords":["constant dimension codes","subspace codes","generalized bilateral multilevel construction","parallel mixed dimension construction","lower bounds","rank-metric codes","Ferrers diagram rank-metric codes","random network coding"],"falsifier":"Take the q=2, n=18, k=4, $\\delta$=2 instance of Example 6, generate one codeword from $C_3$ and one from $\\tilde C_1$, and compute their subspace distance; any pair below 4 would refute Theorem 4. Equivalently, directly check the asserted identity $\\operatorname{wt}(\\bar v'_2)=\\operatorname{wt}(\\bar v_2)+\\operatorname{rank}(\\phi_{\\tilde v}(M))$ on the generator matrix $V=(V_1|V_2)$ obtained by the lifting in that example.","tokens_in":20335,"feed_emoji":"📡","tokens_out":9218,"duration_ms":86886,"temperature":0.7,"pith_summary":"Constant dimension codes are the main tool for error correction in random linear network coding, and the central problem is the exact maximum size $A_q(n,d,\\{k\\})$. This paper tries to establish that the generalized bilateral multilevel construction can be layered on top of the parallel mixed dimension construction, producing constant dimension codes strictly larger than the best previously known ones. The proof works by choosing bilateral identifying vectors of a fixed type whose left part has Hamming weight between $\\delta$ and $k-\\delta$, lifting rank-restricted generalized bilateral Ferrers diagram rank-metric codes, and proving that the union with the parallel construction still has subspace distance at least $2\\delta$. If correct, the paper gives explicit formulas that yield at least 49 new lower bounds, including $A_2(18,4,\\{4\\}) \\ge 5199103860464$.","feed_headline":"Bilateral lifting pushes 49 constant-dimension codes past best-known sizes","feed_subtitle":"The same construction raises the A_2(18,4,{4}) record by more than 2.4 million.","key_machinery":"The carrying object is the generalized bilateral echelon Ferrers form $gEF(\\tilde v)$ and its associated generalized bilateral Ferrers diagram $\\tilde F_{\\tilde v}$, introduced in [19]. A bilateral identifying vector $\\tilde v=(v_1|\\tilde v_3|\\bar v_2)$ splits into a left identifying part and a right inverse-identifying part; the lifted code is built by filling this Ferrers diagram with matrices from a generalized bilateral Ferrers diagram rank-metric (GB-FD) code. The rank condition $\\operatorname{rank}(\\phi_{\\tilde v}(M)) \\le \\operatorname{wt}(v_1)-\\delta$, enforced through Proposition 1, is the mechanism that controls the distance between the new block and the $C_3$ part of the parallel mixed dimension construction; Lemmas 7 and 8 from [19] supply the distance bounds that let the two existing blocks and the new block coexist.","core_discovery":"The paper's central claim is Theorem 4: in the notation of the parallel mixed dimension construction, if $\\tilde S$ is a set of bilateral identifying vectors of type $(\\mu_1,\\mu_3,\\mu_2)$, all of weight $k$ and pairwise Hamming distance at least $2\\delta$, each satisfying $\\delta \\le \\operatorname{wt}(v_1) \\le k-\\delta$, and if for each $\\tilde v \\in \\tilde S$ there is an $(\\tilde F_{\\tilde v},\\delta)_q$ GB-FD code whose codewords satisfy $\\operatorname{rank}(\\phi_{\\tilde v}(M)) \\le \\operatorname{wt}(v_1)-\\delta$, then $C_1 \\cup C_3 \\cup \\tilde C_1$ is an $(n, |C_1|+|C_3|+|\\tilde C_1|, 2\\delta, \\{k\\})_q$ CDC. Theorem 6 turns this into an explicit lower bound on $A_q(n,2\\delta,\\{k\\})$ by taking $\\tilde S$ to be a grid of such vectors and using Proposition 1 to size the GB-FD codes; Theorem 7 gives a second bound based on the corollary of the parallel construction. These bounds beat the previous best-known values in [16] for many parameter sets, for example $A_2(18,4,\\{4\\}) \\ge 5199103860464$.","pith_inferences":["One could iterate the scheme: after forming $C_1\\cup C_3\\cup \\tilde C_1$, regard the enlarged code as a new mixed-dimension ingredient and repeat the bilateral lifting, compounding the gain; the paper does not explore this.","A systematic search over the free parameters $n_1,n_2,n_3,T_1,T_2$ in Theorems 6 and 7 could uncover additional record bounds beyond the 49 reported, since the paper only exhibits selected parameter choices.","The rank restriction $\\operatorname{rank}(\\phi_{\\tilde v}(M)) \\le \\operatorname{wt}(v_1)-\\delta$ is strong; relaxing it to $\\operatorname{wt}(v_1)-\\delta+r$ would trade a controlled amount of minimum distance for larger GB-FD codes, a natural next experiment."],"forward_implications":["Theorem 6 gives a closed-form lower bound for $A_q(n,2\\delta,\\{k\\})$ whenever one has an $(n_1,2\\delta,2\\delta-l_{T_1},T_1)_q$ MDDC and an $(n_3,2\\delta,2\\delta-l_{T_2},T_2)_q$ MDDC; Theorem 7 gives a second bound needing only one MDDC.","Corollaries 2 and 3 provide infinite binary families, $A_2(12+h,4,\\{4\\})$ for $1\\le h\\le5$ and $A_2(18+h,4,\\{4\\})$ for $0\\le h\\le1$.","Corollaries 4 and 5 extend the gains to $q\\ge3$ and to parameter families around $(6\\delta+h,2\\delta,\\{2\\delta\\})$, covering lengths 13 through 19.","At least 49 reported lower bounds strictly improve the best-known values from [16]; for instance the $A_2(18,4,\\{4\\})$ bound grows by 2,413,056 codewords."],"supporting_citations":[{"why":"Supplies the generalized bilateral multilevel construction, GB-FD codes, Proposition 1, and Lemmas 7 and 8 that carry the union argument.","marker":"[19]"},{"why":"Supplies the parallel mixed dimension construction (Theorem 3 and Corollary 1) used as the base code, and the previous best lower bounds used for comparison.","marker":"[16]"},{"why":"Supplies the mixed dimension construction, the MDDC concept, and Theorem 8 used to build the (8,4,3,{4,3})_2 MDDC in the examples.","marker":"[23]"},{"why":"Supplies inverse identifying vectors, the inverse multilevel construction, and Lemmas 4 and 6 used for the distance bound in Case 2.","marker":"[27]"},{"why":"Supplies identifying vectors, Ferrers diagram rank-metric codes, lifting, and Lemmas 3 and 5.","marker":"[6]"},{"why":"Supplies the MRD rank distribution used to compute the sizes of rank-restricted rank-metric codes in the lower-bound formulas.","marker":"[5]"}],"fun_headline_variants":["Bilateral lifting beats 49 best-known constant-dimension code sizes","Generalized bilateral multilevel construction lifts CDC size records","New bound for A_2(18,4,{4}) from bilateral lifting","Pushing 49 CDC sizes past known bounds via bilateral lifting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The union's minimum distance rests on an unproved identity in Case 2 of Theorem 4, namely that the weight of the inverse identifying vector of a bilaterally lifted codeword equals $\\operatorname{wt}(\\bar v_2)+\\operatorname{rank}(\\phi_{\\tilde v}(M))$; if that identity fails, some pair of codewords from the two parts could land closer than $2\\delta$.","fun_headline_variants_meta":{"raw":{"variants":["Bilateral lifting beats 49 best-known constant-dimension code sizes","Generalized bilateral multilevel construction lifts CDC size records","New bound for A_2(18,4,{4}) from bilateral lifting","Pushing 49 CDC sizes past known bounds via bilateral lifting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001352,"raw_usage":{"total_tokens":5506,"prompt_tokens":981,"completion_tokens":4525,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":4450}},"tokens_in":597,"tokens_out":4525,"duration_ms":37391,"temperature":1.0,"reasoning_tokens":4450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:32:08.452956+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the q=2, n=18, k=4, $\\delta$=2 instance of Example 6, generate one codeword from $C_3$ and one from $\\tilde C_1$, and compute their subspace distance; any pair below 4 would refute Theorem 4. Equivalently, directly check the asserted identity $\\operatorname{wt}(\\bar v'_2)=\\operatorname{wt}(\\bar v_2)+\\operatorname{rank}(\\phi_{\\tilde v}(M))$ on the generator matrix $V=(V_1|V_2)$ obtained by the lifting in that example.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the generalized bilateral multilevel construction, GB-FD codes, Proposition 1, and Lemmas 7 and 8 that carry the union argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the parallel mixed dimension construction (Theorem 3 and Corollary 1) used as the base code, and the previous best lower bounds used for comparison."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the mixed dimension construction, the MDDC concept, and Theorem 8 used to build the (8,4,3,{4,3})_2 MDDC in the examples."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies inverse identifying vectors, the inverse multilevel construction, and Lemmas 4 and 6 used for the distance bound in Case 2."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies identifying vectors, Ferrers diagram rank-metric codes, lifting, and Lemmas 3 and 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the MRD rank distribution used to compute the sizes of rank-restricted rank-metric codes in the lower-bound formulas."}],"review_version":1}