{"id":"f76ae350-54ee-4ab6-bc81-760033395c6f","arxiv_id":"1908.03804","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Parallel versions of the linkage construction and of multi-block lifted MRD codes yield new lower bounds on A_q(n,d,k), beating the previous tables in more than 110 cases.","lead":"This paper constructs new constant-dimension subspace codes by running several blocks of rank-metric codes in parallel and counting them with the Delsarte rank distribution. It reports more than 110 improved record sizes for such codes, which are used in error correction for random network coding.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the cross-block intersection bound in Theorem 4.1 is valid, so the central construction and its lower-bound count stand.","rationale":"The reader's weakest assumption correctly identified the cross-block intersection bound in Proposition 4.1 as the key premise of Theorem 4.1. I examined that premise in detail. The apparent risk is that bounding ker(I − AB) by rank(B) can fail if the order of multiplication is reversed; however, the correct row-vector equations put the restricted matrix B = A^j_i on the right in the fixed-point equation α = αAB, so the fixed space is contained in the row space of B and has dimension at most rank(B) ≤ t. This is exactly what the proof needs. I also checked the distinctness of subspaces across block positions, the same-block intersection bound using the MRD property, and the count of codewords block by block; all are consistent with the theorem's formula. Theorem 3.1 was reviewed similarly: the rank condition on Q_2 combined with the nonsingular first block in U gives the needed cross-distance bound, and the two parts of the union are disjoint because the first block is full rank only in W_1. The main mathematical claims therefore appear sound. The remaining weaknesses are not load-bearing for correctness: the missing 'full version' reference, the comparison against [11] rather than the latest tables, and the Added Notes disclosing later extensions in [3,13,19]. These are presentation and novelty concerns that do not invalidate the constructions or the lower bounds, so the reader's conditional verdict should stand unchanged.","tokens_in":15820,"tokens_out":19905,"duration_ms":212624,"concrete_test":"Re-derive Proposition 4.1 with the explicit row-vector substitution: equality at blocks i and j yields α = βA^j_i and β = αA^i_j, hence α = αA^i_jA^j_i; conclude that ker(I−A^i_jA^j_i) ⊆ rowspace(A^j_i), so its dimension is at most rank(A^j_i) ≤ t under the restriction A^j_i ∈ Q_{q,n,t,n-t}. Then verify the bound on a small case by brute force: for q=2, n=2, t=1, s=2, enumerate all 16^2 + 16·A_1 + A_1^2 = 481 subspaces generated by the theorem, compute all pairwise subspace distances, and confirm the minimum is 2(n−t)=2 and the count matches the theorem. If both the analytic substitution and the enumeration pass, the load-bearing cross-block bound is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the reader's nominated weakest assumption, I find it is actually supported. For block positions i<j in Theorem 4.1, equality of vectors in U^i and U^j gives α = βA^j_i (block i) and β = αA^i_j (block j). Substituting gives α = αA^i_j A^j_i, so α lies in the row space of A^j_i. Since A^j_i is restricted to Q_{q,n,t,n-t}, its kernel has dimension at least n−t, hence its rank is at most t. Therefore dim(U^i∩U^j) ≤ rank(A^j_i) ≤ t, giving subspace distance at least 2(n−t). The distinctness claim also holds: equality of subspaces would force A^i_j A^j_i = I, impossible when rank(A^j_i) ≤ t < n. Within a fixed block position, the usual lifted-MRD argument bounds intersections by t because corresponding differences of entries lie in the MRD code Q_{q,n,t}. The counting formula in Theorem 4.1 matches the construction: for block position r, the r−1 entries left of the identity are chosen from Q_{q,n,t,n-t} and the remaining s−r+1 entries from Q_{q,n,t}, giving the factor q^{(s−r+1)n(t+1)} (Σ_{i=n−t}^{t} A_i(Q_{q,n,t}))^{r−1}; summing over r gives the stated bound. I therefore do not find a load-bearing mathematical weakness. The remaining issues are presentational: the unresolved 'full version [ ?]' placeholder, the comparison base [11] rather than the latest tables, and the Added Notes acknowledging later extensions in [3,13,19]. These affect exposition and freshness, not the validity of Theorem 4.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies lower bounds for the maximal size A_q(n,d,k) of constant-dimension subspace codes. It proposes two constructions. The first (Theorem 3.1, Corollary 3.1) is a parallel linkage construction: two families of linked or lifted codes are combined, with one family using rank-restricted subsets of an MRD code as the linking block, and the Delsarte rank distribution is used to count the restricted subset. The second (Theorem 4.1, Corollaries 4.1–4.6) uses s+1 blocks and takes subspaces spanned by rows (A_1,...,I_n,...,A_s), with the identity block in each of the s+1 positions; cross-block intersections are controlled by imposing that the blocks to the left of the identity have rank at most t, and the count is expressed as a sum over the rank distribution of the MRD code Q_{q,n,t}. The constructions yield explicit lower bounds, and the appendix lists many entries claimed to improve the tables in [11].","tokens_in":16155,"tokens_out":11327,"duration_ms":103555,"significance":"The main mathematical content is sound: the distance bound in Theorem 4.1 is valid once the notation is made precise, and the counting formula in that theorem is consistent with the construction. The paper's strength is the systematic use of rank-bounded subsets of MRD codes in several blocks together with the Delsarte rank distribution, which is a clean and reusable idea. The Delsarte-based counting is explicit and can be checked directly from the stated formulas. If the presentation is completed (full tables, resolved placeholders, updated comparison with later work), the paper provides a substantial number of new lower bounds and will be a useful reference for the subspace-codes community.","major_comments":[{"comment":"The notation for the block entries is not defined consistently: the proposition is stated with A_j and B_i, while the proofs of Theorem 4.1 switch to expressions such as A_i^j A_j^i and A_j^i without saying which superscript is the block position and which is the identity position. The load-bearing step is the bound dim(U^i ∩ U^j) ≤ t; as written, the statement that dim ker(I_n − A_i^j A_j^i) ≤ t follows from dim ker(A_j^i) ≥ n−t requires the reader to reconstruct which matrix is the constrained one and to use rank(AB) ≤ rank(B). Proposition 4.1 as stated also asserts the bound for arbitrary A_j and B_i, but the bound to t is true only under the rank constraint on the appropriate block entry. Please define the notation, state the constraint explicitly, and expand the argument, because the minimum distance 2(n−t) of Theorem 4.1 rests entirely on this step.","section":"§4.1 (Proposition 4.1 and Theorem 4.1)"},{"comment":"The manuscript repeatedly refers to 'the full version [ ?]' (for example, §4.2: 'We list all 42 improvements on [11] in Table 2 of the full version [ ?]', and §5: 'Tables 1-5 in the full version [ ?]'). The submitted version contains an appendix with tables, but the unresolved placeholder and the referral to an unidentifiable full version make the claimed 'more than 110 new lower bounds' impossible to verify from the manuscript. Please remove the placeholder, include the complete set of tables or a permanent archive link, and ensure that the count of 110 is supported by the table entries.","section":"§4.2, §5, and Appendix"},{"comment":"The paper compares all new bounds with the 2016 tables [11], but the Added Notes already acknowledge that some results have been extended in [3,13,19]. The manuscript should either update the comparison to the current best-known bounds or explicitly state which of the 110 listed entries remain improvements in the presence of the later work. Without this, the claim that these are 'new constant dimension subspace codes better than previously best known codes' is not substantiated at the time of publication.","section":"§5 (Added Notes)"}],"minor_comments":[{"comment":"The sentence 'We refer the following result to Theorem 5.6 in [5] or Corollary 26 in [4]' should read 'We refer to the following result'.","section":"§2.2"},{"comment":"In the paragraph introducing the subspaces U^j, 'B1,...,Bn are matrices from the MRD code' should be 'B1,...,B_s are matrices from the MRD code'.","section":"§4.1"},{"comment":"The sentence 'From Theorem 4.1 we have the following Corollary 3.2 immediately' should refer to Corollary 4.3, not Corollary 3.2.","section":"§4.2"},{"comment":"The table heading 'A_q(3n. 2(n−t), n)' contains a period where a comma is intended; it should read 'A_q(3n, 2(n−t), n)'.","section":"Appendix, Table 4"},{"comment":"The large integers in the tables are broken across lines without alignment; consider presenting them in a monospaced or comma-separated format so that each entry can be checked reliably against the formulas.","section":"Appendix"}],"recommendation":"major_revision","confidential_remarks":"I did not find a fatal mathematical flaw; the main issues are completeness and self-containedness. The repeated '[?]' full-version references and the comparison base [11] should be resolved before acceptance. The use of the authors' own Corollary 4.1 bound as an input in §3.2 is not circular, but the cross-reference should be cleaned up so that the dependence is transparent. If the authors provide complete tables and update the comparison with [3,13,19], the paper is likely acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's what you should know about arXiv:1908.03804: the main construction is square. I was bracing for the cross-block intersection bound in Theorem 4.1 to be the weak spot, but it holds. For i<j, equality in the row-span representations forces a fixed-point condition that bounds the intersection by t, so the subspace distance is at least 2(n-t), and the Delsarte count gives the stated lower bound. The parallel linkage theorem also checks out. The paper is a genuine, if incremental, contribution to the constant-dimension subspace code lower-bound business.\n\nWhat's new: the arbitrary-s multi-block lifting (Theorem 4.1) and the parallel linkage version (Theorem 3.1). The idea of using bounded-rank subsets of MRD codes, counted by Delsarte's rank distribution, is the key trick. The s=1 case reduces to the authors' own earlier bound from [26], but they cite it and re-derive it via Delsarte, so no circularity. The numerical improvements over the [11] tables are real, roughly 110 parameter sets.\n\nThe soft spots are presentational and minor. Proposition 4.1 is terse; the A^i_j versus A^j_i notation is genuinely confusing and should be cleaned up. The text keeps referring to a 'full version [ ?]' that isn't identified, which is sloppy for an arXiv posting. The comparison base is [11], the 2016 tables; given the Added Notes disclose later extensions in [3,13,19], the claim of 'more than 110 new codes better than previously best known' should be qualified as 'better than the tables in [11]' rather than implying current records. These are copyedit-level issues, not mathematical ones.\n\nFor a specialist in subspace or rank-metric codes, this is worth a serious referee. It won't change the world, but the construction technique is reusable and the bounds are concrete. I'd accept it for peer review with minor revision.","headline":"Sound parallel-block constructions with real lower-bound improvements; the math holds up, and only the presentation needs tightening.","tokens_in":16732,"tokens_out":1966,"would_cite":true,"duration_ms":21016,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B25","94B65","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two constructions that pack bounded-rank slices of MRD codes into parallel blocks prove more than 110 new lower bounds for constant-dimension subspace codes, each improving the best previously known value.","keywords":["constant dimension subspace codes","rank-metric codes","MRD codes","bounded-rank subsets","rank distribution","linkage construction","lifted MRD codes","lower bounds"],"falsifier":"Take a concrete case such as $q=2$, $n=6$, $t=3$, $s=2$ and explicitly compute the intersection of two lifted subspaces from different block positions; any pair with intersection dimension 4 or more refutes the claimed distance $2(n-t)=6$. An independent exact determination of $A_2(18,6,6)$ below the claimed lower bound would also settle the claim.","tokens_in":15580,"feed_emoji":"🧮","tokens_out":18921,"duration_ms":163325,"temperature":0.7,"pith_summary":"This paper targets the core packing question of constant-dimension subspace coding: how many $k$-dimensional subspaces of $\\mathbb{F}_q^n$ can be placed so that any two are at subspace distance at least $d$? It presents two constructions that produce lower bounds for this maximum, $A_q(n,d,k)$, and claims that more than 110 of the resulting bounds are strictly better than the best previously known values. The first construction is a parallel form of the linkage method; the second glues together arbitrary numbers of lifted maximum rank-distance (MRD) codes. In both, the crucial new ingredient is the use of subsets of MRD codes consisting only of matrices of bounded rank, counted exactly by the rank-distribution theorem for MRD codes. If the claims are right, the standard tables of best known subspace-code sizes can be improved in these parameter ranges.","feed_headline":"Parallel MRD blocks yield 110+ new lower bounds","feed_subtitle":"Low-rank slices of rank-metric codes, placed in parallel blocks, beat the best known sizes in 110+ parameter ranges.","key_machinery":"The central objects are the bounded-rank subsets $Q_{q,n,t,k} = \\{f \\in Q_{q,n,t} : \\dim \\ker f \\ge k\\}$ of the MRD code of $q$-polynomial linear maps on $\\mathbb{F}_{q^n}$ with rank distance $n-t$; their sizes are computed from the rank distribution $A_i(Q_{q,n,t})$ of the MRD code. The identity that carries the argument is the cross-block intersection bound of Proposition 4.1: for two lifted row spaces whose identity blocks are in positions $i<j$, the intersection has dimension at most $n - \\mathrm{rank}(I_n - A_j B_i)$, and because the matrix $B_i$ sitting in the earlier block position has rank at most $t$, this dimension is at most $t$. This single estimate turns the disjoint union of parallel lifted MRD codes into a valid constant-dimension subspace code, and the rank-distribution sums then count the codewords.","core_discovery":"The paper's central claim is that bounded-rank subsets of MRD codes can be packed into several parallel blocks without losing distance. Theorem 4.1 states that whenever $2t \\ge n$, for every $s \\ge 1$, $A_q((s+1)n, 2(n-t), n) \\ge \\sum_{j=0}^{s} q^{(s-j)n(t+1)} (\\sum_{i=n-t}^{t} A_i(Q_{q,n,t}))^j$, where $A_i(Q_{q,n,t})$ is the number of rank-$i$ matrices in the $q$-polynomial MRD code $Q_{q,n,t}$. The construction places the $n \\times n$ identity block in each of the $s+1$ positions; for a subspace whose identity lies in block $j$, the matrices in blocks left of the identity are required to have kernel dimension at least $n-t$, i.e., rank at most $t$. That rank restriction makes the intersection of any two subspaces from different block positions have dimension at most $t$, so the subspace distance is at least $2(n-t)$. A parallel linkage construction (Theorem 3.1) gives a similar bound for parameters $(3k+h, d, k)$ with even $d$, and together the two methods yield more than 110 lower bounds better than those in [11].","pith_inferences":["The same mechanism should work with blocks of different sizes or with different MRD codes in different blocks, because the cross-block distance estimate uses only the rank bound on one matrix.","For fixed $n$ and $t$, the relative gain over a single lifted MRD code is largest at small $s$; as $s$ grows, the highest-power term in the sum dominates and the construction approaches the behaviour of one large block.","Because the bounded-rank counts are exact polynomials in $q$, the new bounds can be plugged into the anticode upper bound to measure, for each $q$, how close these parameters come to the theoretical maximum.","A natural testable extension is to relax the condition $2t \\ge n$ by replacing the single low-rank restriction with a two-sided rank restriction, which might allow shorter code lengths."],"forward_implications":["The $s=1$ case of Theorem 4.1 gives $A_q(2n, 2(n-t), n) \\ge q^{n(t+1)} + \\sum_{i=n-t}^{t} A_i(Q_{q,n,t})$, producing 42 entries in the paper's tables that improve previous records.","Applying the Johnson-type bound to the $s=1$ construction yields 7 improved lower bounds for parameters of the form $A_q(2n-1, 2(n-t), n-1)$, including $A_2(17,6,8)$.","The parallel linkage construction of Theorem 3.1 and Corollary 3.1 gives 63 improved bounds for $A_q(3k+h, d, k)$ with even $d$, such as $A_2(18,6,6)$ and $A_2(19,6,6)$.","The $s=2$ and $s=3$ cases of the multi-block construction produce new lower bounds, including values for $A_2(20,4,5)$ and $A_2(24,6,6)$ that cannot be compared with any entry in [11].","In total, the two methods yield more than 110 constant-dimension subspace codes whose sizes exceed the best previously known lower bounds."],"supporting_citations":[{"why":"Supplies the rank-distribution formula for MRD codes used in every count of bounded-rank subsets.","marker":"[5]"},{"why":"Provides the $q$-polynomial MRD code $Q_{q,n,t}$, its rank distance $n-t$, and its size $q^{n(t+1)}$.","marker":"[9]"},{"why":"The linkage construction that Theorem 3.1 runs in parallel, preserving subspace distance under concatenation.","marker":"[10]"},{"why":"Tables of best known constant-dimension subspace code sizes that all 110+ new bounds are compared against and improve.","marker":"[11]"},{"why":"Gives the lifted MRD code construction, the building block for the multi-block method.","marker":"[22]"},{"why":"The Johnson-type bound used in Corollary 4.2 to convert even-length bounds into odd-length ones.","marker":"[8]"}],"fun_headline_variants":["Subspace codes from MRD subsets in parallel blocks beat 110 records","Bounded-rank MRD slices yield 110+ new subspace code lower bounds","Parallel blocks of bounded MRD ranks set 110+ new subspace code sizes","MRD low-rank blocks boost subspace codes past 110 best-known bounds","Subspace codes via MRD subsets: 110+ improved lower bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction's distance guarantee rests entirely on the estimate that a rank-at-most-$t$ matrix in an earlier block keeps the intersection of two lifted subspaces to dimension at most $t$; if that estimate ever failed, the minimum subspace distance $2(n-t)$ would drop and the claimed code would not exist.","fun_headline_variants_meta":{"raw":{"variants":["Subspace codes from MRD subsets in parallel blocks beat 110 records","Bounded-rank MRD slices yield 110+ new subspace code lower bounds","Parallel blocks of bounded MRD ranks set 110+ new subspace code sizes","MRD low-rank blocks boost subspace codes past 110 best-known bounds","Subspace codes via MRD subsets: 110+ improved lower bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000966,"raw_usage":{"total_tokens":4138,"prompt_tokens":1000,"completion_tokens":3138,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":3040}},"tokens_in":616,"tokens_out":3138,"duration_ms":23117,"temperature":1.0,"reasoning_tokens":3040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:02:58.598878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete case such as $q=2$, $n=6$, $t=3$, $s=2$ and explicitly compute the intersection of two lifted subspaces from different block positions; any pair with intersection dimension 4 or more refutes the claimed distance $2(n-t)=6$. An independent exact determination of $A_2(18,6,6)$ below the claimed lower bound would also settle the claim.","supporting_citations":[{"cited_title":"Delsarte, Bilinear forms over a finite field, with appl ications to coding theory, Journal of Combinatorial Theory, Series A, v ol","cited_arxiv_id":null,"evidence_quote":"Supplies the rank-distribution formula for MRD codes used in every count of bounded-rank subsets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $q$-polynomial MRD code $Q_{q,n,t}$, its rank distance $n-t$, and its size $q^{n(t+1)}$."},{"cited_title":"Gluesing-Luerssen and C","cited_arxiv_id":null,"evidence_quote":"The linkage construction that Theorem 3.1 runs in parallel, preserving subspace distance under concatenation."},{"cited_title":"Silva, F","cited_arxiv_id":null,"evidence_quote":"Gives the lifted MRD code construction, the building block for the multi-block method."},{"cited_title":"Etzion and A","cited_arxiv_id":null,"evidence_quote":"The Johnson-type bound used in Corollary 4.2 to convert even-length bounds into odd-length ones."}],"review_version":1}