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REVIEW 2 major objections 6 minor 40 references

Asymptotically optimal cyclic subspace codes

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict New nesting operation for cyclic subspace codes; solid construction with an overclaim in the abstract that should be fixed. read the letter →

arxiv 2507.09290 v1 pith:AKTX5PYP submitted 2025-07-12 cs.IT math.COmath.IT

Chiara Castello , Paolo Santonastaso This is my paper · ORCID
classification cs.ITmath.COmath.IT MSC 11T7111T9994B99
keywords subspacecodescyclicSidonspacesGrassmannianJohnsonboundrandomnetworkcodingfinitefieldsconstant-dimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $n/k$.

What carries the argument

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 $\alpha\in\mathbb{F}_{q^n}\setminus\mathbb{F}_q$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [Abstract and Section 1] 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.
  2. [Theorem 3.8, Theorem 3.10, and Theorem 4.6] 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.
minor comments (6)
  1. [Section 1] In the first paragraph, 'refereed' should be 'referred'.
  2. [Example 3.12] 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.
  3. [Theorem 4.1] 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.
  4. [Example 2.18] 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}}.
  5. [Equation (32)] 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.
  6. [Construction 3.9] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nesting machinery is proved internally and the base-code inputs are independent external theorems, not self-citations or fitted predictions.

full rationale

The paper's derivation chain separates cleanly into an internal nesting operation and externally cited base codes. The operation ⊙ is defined in Definitions 2.4 and 2.8; its size and distance behavior are proved in Theorem 2.14 and Theorem 2.19 directly from the injectivity of the maps Φ and from intersection bounds, without assuming the conclusion. Corollary 2.15 and Theorem 2.19 then multiply cardinalities and preserve minimum distance under the stated hypotheses. The special constructions in Section 3 instantiate this machinery with base codes that are explicitly cited from prior work by other authors: Theorem 3.2 cites [24, Lemma 38] for the 2C_k base codes, and Theorem 3.8 states "see [35, Theorem 3.5]" for the p C_k base codes. Theorems 3.4, 3.10, 3.15 and 3.18 apply Theorem 2.19 to these external building blocks, and the asymptotic comparisons in Theorems 4.4, 4.5 and 4.6 are direct estimates combining the resulting product formulas with the Johnson type bound II from (36). No equation reduces to its own input by construction: the size formulas are products of the cited base-code sizes, and the minimum distance claims follow from the independently stated distances of the base codes plus the internal Theorem 2.14 argument. The self-citations in the paper (e.g., [3], [4], [5], [23]) appear in contextual remarks or in the final optical-orthogonal-code observation and are not load-bearing for the central asymptotic-optimality claim. The noted discrepancy between the abstract's phrasing and Remark 4.2 (the r = 4 case only reaches a 1/4 factor of the Johnson bound) is an overstatement in presentation, not a circular step. The main correctness risk is the dependence of the tower on the unverified-in-this-paper external base theorem [35, Theorem 3.5] for p = 3, but a dependence on an external independent theorem is not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The construction introduces no free fitted parameters and no new postulated entities. It relies on standard finite field theory and on the cited base constructions of Sidon spaces, which provide the building blocks for the tower. The central claim is carried by the new operation odot and the recursion theorems, not by any parameter fitted to data.

assumptions (5)
  • standard math Sidon space characterization: Orb(U) has minimum distance 2k-2 and size (q^n-1)/(q-1) iff dim(U ∩ alpha U) <= 1 for alpha not in F_q (Theorem 2.11, citing [24, Lemma 34]).
    Used to show that base codes have minimum distance 2k-2 and full length, and to guarantee the representative maps have the needed properties.
  • standard math Base code [24, Construction 37 and Lemma 38]: for n=2k and q at least 3, there exist tau = floor((q-1)/2) Sidon spaces whose orbits are distinct, and each code 2C_k has size tau (q^{2k}-1)/(q-1) and minimum distance 2k-2.
    This is the foundation for the 2-power tower constructions in Construction 3.3 and Theorem 3.4.
  • standard math Base code [35, Theorem 3.5]: for an odd prime p, the code pC_k has size q^k (q^{(ell+1)k}-1)(q^{pk}-1)/(q^k-1) and minimum distance 2k-2, with representatives given as images of injective Fq-linear maps pPhi_{k,h}.
    This is the foundation for the odd-prime tower constructions in Construction 3.9 and Theorem 3.10.
  • standard math Stabilizer cardinality lemma [22, Theorem 1]: Orb(V) has cardinality (q^n-1)/(q^t-1) if and only if Stab(V) = F*_{q^t}.
    Used in Lemma 2.5 and Proposition 2.6 to establish full-length properties of the nested codes.
  • standard math Standard finite field facts: norm maps between finite fields are surjective, and primitive elements exist in every finite field, so the elements omega_i and gamma_i required in the constructions exist.
    These existence facts are invoked implicitly in Constructions 3.3, 3.9, 3.14 and 3.17.

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Cite this review

Pith. "Pith review of Asymptotically optimal cyclic subspace codes." pith.science (2026). https://pith.science/paper/AKTX5PYP

@misc{pith2026250709290,
  author       = {Pith},
  title        = {Pith review of: Asymptotically optimal cyclic subspace codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKTX5PYP}},
  note         = {Machine review of arXiv:2507.09290}
}
abstract

Subspace codes, and in particular cyclic subspace codes, have gained significant attention in recent years due to their applications in error correction for random network coding. In this paper, we introduce a new technique for constructing cyclic subspace codes with large cardinality and prescribed minimum distance. Using this new method, we provide new constructions of cyclic subspace codes in the Grassmannian $\mathcal{G}_q(n,k)$ of all $k$-dimensional $\mathbb{F}_q$-subspaces of an $n$-dimensional vector space over $\mathbb{F}_q$, when $k\mid n$ and $n/k$ is a composite number, with minimum distance $2k-2$ and large size. We prove that the resulting codes have sizes larger than those obtained from previously known constructions with the same parameters. Furthermore, we show that our constructions of cyclic subspace codes asymptotically reach the Johnson type bound II for infinite values of $n/k$.

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