{"id":"7c188a50-63f4-43a8-90b2-f9c2d1ad2a9a","arxiv_id":"2507.09290","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A nesting operation builds cyclic subspace codes with minimum distance 2k-2 whose size asymptotically matches the Johnson bound when n/k is a power of 3.","lead":"Cyclic subspace codes are error-correcting codes for network coding, and this paper shows how to build larger ones by nesting codes defined on smaller fields into codes on larger fields. For dimension ratios that are powers of 3, the new codes are the first cyclic codes whose size provably reaches the best known upper bound as the field or dimension grows.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central asymptotic-optimality claim for 3^e C_k (Theorem 4.6) rests entirely on the unverified p=3 base code cited from [35, Theorem 3.5]; if that base code fails for some q,k, the tower construction collapses.","rationale":"The reader's CONDITIONAL verdict is justified, and the external base-code dependency is the same concern I identify, but I narrow it: the central claim in Theorem 4.6 only needs [35, Theorem 3.5] for p=3, not the full panoply of base codes. The 2^e family only reaches a constant factor 2^e from the bound, so errors there would not change the asymptotic-optimality claim. I checked the internal proof chain: Theorem 2.14's distinctness argument is sound, the induction in Theorem 2.19 applies cleanly, and the exponent calculation in (30) is correct. The overclaim against previous constructions (Remark 4.2) is real but not load-bearing for the optimality result. Hence the decisive risk is the unproved p=3 base code. The concrete test checks the smallest case where the orbit count and minimum distance are nontrivial; a counterexample would immediately invalidate Theorem 4.6, and a success would motivate a full analytic re-derivation of [35].","tokens_in":31531,"tokens_out":20126,"duration_ms":211915,"concrete_test":"Exhaustively verify the p=3 base code for the smallest nontrivial parameters. For q=2, k=2, choose any gamma in F_64 excluding F_4 and enumerate, in G_q(6,2), the union C = union over a in F_4 of Orb({u + (u^2 + a u) gamma : u in F_4}), with j=0 only since q-2=0. Compute the number of distinct codewords and the minimum subspace distance. Accept the base code only if |C|=252 and d(C)=2. If either fails, Theorem 4.6 is unsupported; if it passes, a more decisive step is to re-derive [35, Theorem 3.5] analytically for p=3, since a single small-case success does not prove all q,k.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the base case of the tower: Construction 3.9 builds 3^e C_k by nesting e copies of the code 3C_{3^{i-1}k} defined in Construction 3.7, and Theorem 3.10 multiplies their sizes and preserves minimum distance. The internal nesting machinery (Theorem 2.14 and its induction Theorem 2.19) is proved in the paper and appears sound; the distinctness argument via injective maps and intersection bounds is clean. Therefore the only unproved input that Theorem 4.6 needs is Theorem 3.8, which is exactly [35, Theorem 3.5] for the special case p=3: for every k at least 2 and every prime power q, the union of the q^k(q-1) orbits of the subspaces u maps to u + omega^j(u^q + a u)gamma (with a in F_q^k, j from 0 to q-2, gamma generating F_q^{3k} over F_q^k) must have exactly q^k(q^{3k}-1) codewords and minimum distance 2k-2. No proof is given in this paper; it is an external citation. If the size or the distance is off for even one pair (q,k), the product formula in Theorem 3.10, the asymptotic exponent in (30), and the ratio tending to 1 in Theorem 4.6 all fail. The paper's own Remark 4.2 shows that the abstract overclaim is false for r=4, but that is a presentation issue; the mathematically load-bearing risk is the unverified base code, not the overclaim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an operation ⊙ that, given cyclic subspace codes C1 ⊆ G_q(m,k) and C2 ⊆ G_q(n,m), produces a cyclic subspace code C2 ⊙ C1 ⊆ G_q(n,k). Theorems 2.14 and 2.19 give conditions under which the minimum distance remains 2k−2 and the cardinality multiplies: |C2 ⊙ C1| = |C2||C1|. The authors apply this operation to known base constructions, one for r=2 from [24] and one for odd primes p from [35], to build codes in G_q(rk,k) for composite r. The main results are Theorem 3.4 for r=2^e, Theorem 3.10 for r=p^e, Theorem 3.15 for products of odd prime powers, Theorem 3.18 for r=2^e times an odd part, and Theorem 4.6, which claims that the 3^e family is asymptotically optimal both in k and in q. The paper also compares the new sizes with the Johnson type bound II and with the earlier constructions summarized in Table 1.","tokens_in":31796,"tokens_out":9673,"duration_ms":103240,"significance":"If the claims hold, the nesting technique is a genuine and potentially influential contribution: it gives explicit families of cyclic subspace codes with minimum distance 2k−2 whose sizes asymptotically meet the Johnson type bound II for infinitely many values of n/k, specifically r=3^e with e≥2, both as k→∞ and as q→∞. The proof of Theorem 2.14 via pairwise intersection bounds is careful, and the induction in Theorem 2.19 is sound. The new constructions also provide concrete improvements over the previous state of the art for composite r≥6. The principal caveat is that the headline asymptotic-optimality result depends entirely on the external base code for p=3 imported as Theorem 3.8 from [35], which is not proved in the paper; the abstract also overclaims the range of parameters for which the new codes beat all previous constructions.","major_comments":[{"comment":"The abstract and introduction state that the new codes have sizes larger than those of all previously known constructions whenever k|n and n/k is composite, but this is not what the theorems prove. Theorem 4.1 restricts the comparison to composite r≥6, and Remark 4.2 explicitly concedes that for r=4 the new code satisfies |2^2 C_k| ∼ (1/4)q^{6k}, while Construction 2 of Table 1 has S2(4k,k,q) ∼ (1/2)q^{6k}. Thus the new code is asymptotically half the size of the best known code for r=4. The abstract and introduction should be rewritten to state the parameter range accurately, e.g. composite r≥6, with the separate r=4 and r=2^e cases described only by their factor-1/2^e behavior relative to the Johnson bound.","section":"Abstract and Section 1"},{"comment":"The asymptotic-optimality claim in Theorem 4.6 rests on the p=3 base case, which is quoted as Theorem 3.8 from [35, Theorem 3.5] and is not proved in the paper. The nesting machinery in Theorems 2.14 and 2.19 is internal and appears sound, but the product formula in Theorem 3.10, the asymptotic exponent in (30), and the ratio tending to 1 in Theorem 4.6 all depend on the exact cardinality and minimum distance of the base code 3C_k, including the fact that the q^k(q−1) orbits are distinct for every k≥2 and every prime power q. The authors should either provide a self-contained proof of this p=3 special case, or quote the precise theorem with all hypotheses and verify explicitly that those hypotheses are satisfied for every parameter used in the paper. Without this, the main optimality result is conditional on an unverified external input.","section":"Theorem 3.8, Theorem 3.10, and Theorem 4.6"}],"minor_comments":[{"comment":"In the first paragraph, 'refereed' should be 'referred'.","section":"Section 1"},{"comment":"The three codes are listed as 3C_k, 3C_{9k}, and 3C_{9k}; the second should be 3C_{3k} so that the final nesting 3C_{9k} ⊙ 3C_{3k} ⊙ 3C_k is defined consistently.","section":"Example 3.12"},{"comment":"In the odd-r case, the q-limit statement reads 'S3(rk,k,q), S5(rk,k,q) = o_q(|rCk|)'; since S5 is repeated, the intended statement is presumably 'S3 and S4', matching the k-limit sentence. Please correct the typo and make the treatment of S4 and S5 explicit in the proof.","section":"Theorem 4.1"},{"comment":"The summation variable ξ should range over F^*_{q^{27k}}, not F_{q^{27k}}, because the code is an orbit under the multiplicative group of F_{q^{27k}}.","section":"Example 2.18"},{"comment":"The exponent contains a typesetting artifact '2 eQ_{i=1}^e'; it should be '2 \\prod_{i=1}^e p_i^{e_i} k (2^e−1)' or similar. Please fix the display.","section":"Equation (32)"},{"comment":"In the displayed definition of pV_{p^{i-1}k,h_i}, the variables u and v are mixed inside the same expression; use a single variable consistently.","section":"Construction 3.9"}],"recommendation":"major_revision","confidential_remarks":"The paper's internal nesting construction is sound and the contribution is potentially significant, but the abstract overclaim for r=4 and the complete dependence of Theorem 4.6 on the unproved p=3 base case imported from [35] should be addressed before acceptance. I would not reject the paper merely for relying on a published theorem; however, the manuscript currently presents the asymptotic-optimality result as self-contained when it is not, and the advertised range of parameters is false as written. Asking the authors to prove or precisely quote the p=3 base case and to correct the abstract is a reasonable path to a publishable paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Castello–Santonastaso paper on cyclic subspace codes. The one thing to know: the new ⊙-operation is genuinely new and does what it says—given two cyclic codes in a tower of field extensions, it produces a code in the larger Grassmannian with size the product of the two and the same relative minimum distance. The proofs of Theorems 2.14 and 2.19 are clean, and the induction multiplies correctly. That gives the first infinite families asymptotically attaining the Johnson type bound II for n/k = 3^e, e ≥ 2, which is a real step beyond the known r=3 case. The paper deserves a serious referee.\n\nWhat is good specifically: the size counting via pairwise intersection bounds is careful; the distinctness argument using injective maps is correct; the asymptotic comparisons in Section 4 are straightforward and check out. The authors also correctly note in Remark 4.3 that previous constructions only meet the bound for r=3.\n\nWhere I'd push back: the abstract and introduction overclaim. The abstract says the new codes are \"larger than all previously known constructions with the same parameters\" for all composite n/k. That is false for r=4, as their own Remark 4.2 shows—for r=4 the known code from [31] has size ~(1/2)q^{6k}, while their 4C_k is ~(1/4)q^{6k}. This should be fixed before publication, and the intro/conclusion should be trimmed to match Theorem 4.1, which restricts to r ≥ 6.\n\nSecond, the stress-test concern: the main theorem for 3^e C_k rests on the unproved base code p=3, quoted from [35, Theorem 3.5]. That is a published theorem, so it is not a fatal flaw, but the authors would make the paper safer by including a short verification of the p=3 case rather than citing it wholesale. The dependency is explicit and the tower collapses if the base is wrong—that is true of any nesting argument, but here it is worth stating clearly. I would call this minor-to-moderate, not a reason to reject.\n\nAlso, the conclusion has a slipped phrase saying r=3^e is \"far from\" the bound; it is their optimal case. That is a typo, not a mathematical issue.\n\nBottom line: send it to peer review. The core construction is sound, the overclaim is fixable, and the p=3 base should be checked or proved inline. If the base holds (very likely, it is published), the paper is a strong contribution to subspace codes.","headline":"New nesting operation for cyclic subspace codes; solid construction with an overclaim in the abstract that should be fixed.","tokens_in":32403,"tokens_out":4824,"would_cite":true,"duration_ms":52212,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11T71","11T99","94B99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Iterating a nesting operation builds cyclic subspace codes whose size asymptotically reaches the Johnson type bound II whenever the ratio n/k is a power of 3.","keywords":["subspace codes","cyclic subspace codes","Sidon spaces","Grassmannian","Johnson bound","random network coding","finite fields","constant-dimension codes"],"falsifier":"For $q=2$, $k=2$, $p=3$, take $\\gamma\\in\\mathbb{F}_{64}$ with $\\mathbb{F}_4(\\gamma)=\\mathbb{F}_{64}$ and form the four representatives $V_a=\\{u+(u^2+a u)\\gamma:u\\in\\mathbb{F}_4\\}$, $a\\in\\mathbb{F}_4$; if their orbit union contains fewer than $4\\cdot 63$ distinct codewords, or any two distinct codewords meet in a space of dimension $2$ (subspace distance below $2$), then the base code $3C_2$ does not have the claimed parameters and the asymptotic-optimality tower fails.","tokens_in":31245,"feed_emoji":"🔢","tokens_out":12272,"duration_ms":133134,"temperature":0.7,"pith_summary":"The paper introduces a way to combine cyclic subspace codes living in different Grassmannians, nesting a code in $\\mathcal{G}_q(m,k)$ inside a code in $\\mathcal{G}_q(n,m)$ to produce a code in $\\mathcal{G}_q(n,k)$. When both ingredients have the best possible minimum distance for their dimensions, namely $2k-2$ and $2m-2$, the resulting code has that same best distance and its size multiplies. Iterating this over a tower of field extensions gives cyclic subspace codes in $\\mathcal{G}_q(rk,k)$ for every composite $r$, with minimum distance $2k-2$ and sizes larger than all previously known codes with these parameters. For $r=3^e$ the size is asymptotically $q^{2(n-k)}$, which is exactly the asymptotic value of the Johnson type bound II, so these are the first cyclic subspace codes shown to be asymptotically optimal for infinitely many ratios $n/k$.","feed_headline":"Subspace codes reach the Johnson bound for ratios 3^e","feed_subtitle":"A nesting operation multiplies code sizes while preserving minimum distance, yielding asymptotically optimal cyclic codes.","key_machinery":"The load-bearing object is the nesting operation $\\odot$. For an injective $\\mathbb{F}_q$-linear map $\\Phi:\\mathbb{F}_{q^m}\\to\\mathbb{F}_{q^n}$ with image $V_2$, and a subspace $V_1\\subseteq\\mathbb{F}_{q^m}$, it sets $V_2\\odot V_1=\\Phi(V_1)$, then extends to orbits and unions of orbits. Theorem 2.14 shows that if $C_1\\subseteq\\mathcal{G}_q(m,k)$ has minimum distance $2k-2\\ell$ and $C_2\\subseteq\\mathcal{G}_q(n,m)$ has minimum distance $2m-2\\ell'$, then $C_2\\odot C_1$ has minimum distance at least $2k-2\\max\\{\\ell,\\ell'\\}$, and when this maximum is below $k$ its size is $|C_2||C_1|$; in the best-distance case $\\ell=\\ell'=1$, the distance $2k-2$ is preserved. Iterating via Theorem 2.19 multiplies the sizes of all layers. The proof that the base one-orbit codes have distance $2k-2$ uses the Sidon-space criterion: such an orbit has this distance exactly when its representative $U$ satisfies $\\dim(U\\cap\\alpha U)\\le 1$ for every $\\alpha\\in\\mathbb{F}_{q^n}\\setminus\\mathbb{F}_q$.","core_discovery":"The central result is Theorem 4.6: for every $e\\ge 2$ and every prime power $q$, the code $3^e C_k$ obtained by nesting the base construction for ratio $3$ a total of $e$ times has minimum distance $2k-2$ and cardinality asymptotically equal to $q^{2(3^e-1)k}=q^{2(n-k)}$ as $k$ or $q$ tends to infinity, matching the Johnson type bound II, $J_q(n,2k-2,k)\\sim q^{2(n-k)}$. Hence these codes are asymptotically optimal in both $k$ and $q$. The same machinery gives codes for $n/k=2^e$ and $n/k=2^e3^{e_1}$ whose sizes lie within a factor $1/2^e+o(1)$ of the Johnson bound, and for every composite $r\\ge 6$ the new code $rC_k$ is asymptotically larger than each of the previously known cyclic constructions with the same parameters.","pith_inferences":["Editorial: the $1/2^e$ loss for powers of $2$ comes entirely from the base code with ratio $2$, whose size carries the factor $\\lfloor(q-1)/2\\rfloor$; a base construction for ratio $2$ with a full $(q^k-1)$ factor would remove that loss and make every $2^e$-ratio tower asymptotically optimal.","Editorial: the exponent calculation shows that among odd primes $p$, the cited base code $pC_k$ reaches the Johnson bound asymptotically only for $p=3$; building an equally strong base code for another prime $p$ would, by the tower argument, give asymptotically optimal cyclic codes for every composite $r$ made from that prime.","Editorial: because $\\odot$ is defined on subspaces rather than coordinates, a natural testable extension is whether the tower idea survives when $k\\nmid n$, a case the paper leaves open; any nesting using injective maps between extensions of unequal degrees would be a first step in that direction."],"forward_implications":["For every $e\\ge 2$ and every $q$, there exist cyclic subspace codes in $\\mathcal{G}_q(3^e k,k)$ with minimum distance $2k-2$ and size asymptotic to the Johnson type bound II, so asymptotically optimal cyclic codes exist for infinitely many ratios $n/k$.","For $n/k=2^e$ and for $n/k=2^e3^{e_1}$ with $q>2$, the new codes lie within the factor $1/2^e+o(1)$ of the Johnson bound as $k$ or $q$ grows, improving on the previous $1/2+o(1)$ best behavior for these parameters.","Because sizes multiply exactly across layers, the asymptotic exponent of the final code is the sum of the exponents of the layers; this product structure is what drives the comparison against all earlier constructions.","As the authors note at the end of the paper, the new multi-orbit cyclic subspace codes also yield new families of optical orthogonal codes with new parameters."],"supporting_citations":[{"why":"Supplies the base code $2C_k$ with minimum distance $2k-2$ and cardinality $\\tau(q^{2k}-1)/(q-1)$, together with the Sidon-space criterion used to identify best-distance one-orbit codes.","marker":"[24]"},{"why":"Supplies the base code $pC_k$ for each odd prime $p$, with its stated size and minimum distance $2k-2$, plus the injective linear maps that form each layer of the nesting.","marker":"[35]"},{"why":"Supplies the stabilizer lemma linking an orbit's size to the stabilizer of its representative, used to prove the nested codes are full-length.","marker":"[22]"},{"why":"Supplies the Johnson type bound II, the upper-bound benchmark against which asymptotic optimality is measured.","marker":"[30]"},{"why":"Supplies the cyclic subspace code framework and the Johnson-type bound formulation used in the comparison.","marker":"[7]"}],"fun_headline_variants":["Nested cyclic codes hit Johnson bound for 3^e ratios","Cyclic subspace codes reach Johnson bound asymptotically","Nesting yields asymptotically optimal cyclic codes","Composite-ratio cyclic codes meet Johnson bound","New construction matches Johnson bound for composite ratios"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole tower rests on the cited base constructions: if for some $q$ or $k$ a building-block code with ratio $2$ or ratio an odd prime $p$ fails to have exactly the claimed size and minimum distance $2k-2$, then the product-size formulas and the asymptotic-optimality theorems collapse.","fun_headline_variants_meta":{"raw":{"variants":["Nested cyclic codes hit Johnson bound for 3^e ratios","Cyclic subspace codes reach Johnson bound asymptotically","Nesting yields asymptotically optimal cyclic codes","Composite-ratio cyclic codes meet Johnson bound","New construction matches Johnson bound for composite ratios"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000357,"raw_usage":{"total_tokens":1921,"prompt_tokens":917,"completion_tokens":1004,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":932}},"tokens_in":533,"tokens_out":1004,"duration_ms":9099,"temperature":1.0,"reasoning_tokens":932,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:02:09.390885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $q=2$, $k=2$, $p=3$, take $\\gamma\\in\\mathbb{F}_{64}$ with $\\mathbb{F}_4(\\gamma)=\\mathbb{F}_{64}$ and form the four representatives $V_a=\\{u+(u^2+a u)\\gamma:u\\in\\mathbb{F}_4\\}$, $a\\in\\mathbb{F}_4$; if their orbit union contains fewer than $4\\cdot 63$ distinct codewords, or any two distinct codewords meet in a space of dimension $2$ (subspace distance below $2$), then the base code $3C_2$ does not have the claimed parameters and the asymptotic-optimality tower fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the base code $2C_k$ with minimum distance $2k-2$ and cardinality $\\tau(q^{2k}-1)/(q-1)$, together with the Sidon-space criterion used to identify best-distance one-orbit codes."},{"cited_title":"Zhang, C","cited_arxiv_id":null,"evidence_quote":"Supplies the base code $pC_k$ for each odd prime $p$, with its stated size and minimum distance $2k-2$, plus the injective linear maps that form each layer of the nesting."},{"cited_title":"Otal and F","cited_arxiv_id":null,"evidence_quote":"Supplies the stabilizer lemma linking an orbit's size to the stabilizer of its representative, used to prove the nested codes are full-length."},{"cited_title":"Xia and F.-W","cited_arxiv_id":null,"evidence_quote":"Supplies the Johnson type bound II, the upper-bound benchmark against which asymptotic optimality is measured."},{"cited_title":"Etzion and A","cited_arxiv_id":null,"evidence_quote":"Supplies the cyclic subspace code framework and the Johnson-type bound formulation used in the comparison."}],"review_version":1}