Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction

Fang-Wei Fu; Han Li

REVIEW 2 major objections 4 minor 36 references

Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.07842 v1 pith:3WAUMC6K submitted 2025-07-10 cs.IT math.IT

Han Li , Fang-Wei Fu This is my paper · ORCID

classification cs.ITmath.IT MSC 94B6094B65

keywords constantdimensioncodessubspacegeneralizedbilateralmultilevelconstructionparallelmixedlowerboundsrank-metricFerrersdiagramrandomnetworkcoding

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

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 $A_2(18,4,\{4\}) \ge 5199103860464$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $A_2(18,4,\{4\}) \ge 5199103860464$.

Load-bearing premise

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 $2\delta$.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

The 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[§3, Theorem 5] 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′.
[§3, proof of Theorem 4, Case 2] 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.

minor comments (4)

[Example 2] 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.
[Proof of Theorem 6] 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.
[Proof of Theorem 4, Case 1] 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.
[Application of Proposition 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the new lower bounds are explicit additive combinations of prior constructions with no fitted parameters or target values assumed.

full rationale

The paper's derivation chain is additive and non-circular. Theorems 4 and 5 prove that the generalized bilateral multilevel code tilde C1 (or tilde C1') can be unioned with the parallel mixed dimension code C1 union C3 (or C1 union C3'), and the new size is the sum |C1| + |C3| + |tilde C1|. The improvement over [16] comes from an explicit extra term, e.g. |tilde C1| = 2413056 in Example 6, computed by counting GB-FD codes through Proposition 1; it is not obtained by assuming the target lower bound. No parameter is fitted to the old bounds, and no 'prediction' is a renamed input: the old value 5199101447408 appears only as the comparison baseline, while the new value 5199103860464 is the old value plus a separately derived explicit sum. Theorems 6 and 7 are direct size formulas from MRD/RRD and GB-FD code counts, so the resulting lower bounds are self-contained consequences of previously established constructions. The only substantive weaknesses are correctness gaps, not circularity: Theorem 5's proof is omitted ('The proof is similar to that of Theorem 4 and is omitted here'), and Theorem 4 Case 2 asserts the identity wt(bar v'_2) = wt(bar v2) + rank(phi_tilde v(M)) without proof. If that identity failed, the distance guarantee for C1 union C3 union tilde C1 would fail, making the Theorem 6 bounds conditional; likewise Theorem 7 depends on the omitted proof of Theorem 5. These are proof-completeness risks and should be resolved by supplying the missing arguments, but they do not make the derivation circular. There is also no load-bearing self-citation chain: the cited constructions [16] and [19] are by other author groups, not by Li and Fu, and the paper's central union argument is carried by its own distance estimates. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central construction inherits all its building blocks from prior work [16, 19, 23]. No free parameters are fitted; the choices of identifying vectors are explicit functions of n, k, and δ. The paper contributes the selection criteria and the resulting size formulas, not new axioms.

assumptions (6)

standard math Existence of MRD codes for all feasible parameters (Delsarte, Gabidulin)
Used in Theorems 2, 3, 6, and 7 to provide rank-metric codes P_t and Q_s; standard theorem cited as [5, 9, 31].
domain assumption Optimal FD codes from Lemma 2 (Etzion et al.)
Used implicitly in multilevel constructions; cited as [8].
domain assumption Proposition 1 on sizes of rank-restricted GB-FD codes
Core ingredient for the size of the added bilateral code set tilde C1; taken from [19], not proved in this paper.
domain assumption Lemmas 7 and 8 on subspace distances for generalized bilateral echelon forms
Justify the distance bounds for the new code set; from [19].
domain assumption Theorem 8, existence of MDDCs with prescribed dimension distribution
Used to build the mixed dimension input codes X1 and X3 in Corollaries 4 and 5; from [23].
domain assumption The (8,4,3,{4,3})_2 MDDC with eta4 = 4801 and eta3 = 327 exists
Used in Examples 6 and 7 and in Corollaries 2 and 3; from [2, 23].

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Cite this review

Pith. "Pith review of Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction." pith.science (2026). https://pith.science/paper/3WAUMC6K

@misc{pith2026250707842,
  author       = {Pith},
  title        = {Pith review of: Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3WAUMC6K}},
  note         = {Machine review of arXiv:2507.07842}
}

abstract

Constant dimension codes (CDCs), as special subspace codes, have received extensive attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximal possible size $A_q(n,d,\{k\})$ for given parameters $q, n, d$, and $k$. This paper introduces criteria for choosing appropriate bilateral identifying vectors compatible with the parallel mixed dimension construction (Des. Codes Cryptogr. 93(1):227--241, 2025). We then utilize the generalized bilateral multilevel construction (Des. Codes Cryptogr. 93(1):197--225, 2025) to improve the parallel mixed dimension construction efficiently. Many new CDCs that are better than the previously best-known codes are constructed.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [16]

He X., Zhang Z., Tian S., Wang J., Chen Y.: Parallel construction for constant dimension codes from mixed dimension construction. Des. Codes Cryptogr. 93(1), 227–241 (2025)

work page 2025

[2] [1]

IEEE Trans

Antrobus J., Gluesing-Luerssen H.: Maximal Ferrers diagram codes: constructions and genericity considerations. IEEE Trans. Inf. Theory 65(10), 6204–6223 (2019)

work page 2019

[3] [2]

Braun M., ¨Osterg ˚ ard P.R.J., Wassermann A.: New lower bounds for binary constant- dimension subspace codes. Exp. Math. 27(2), 179-183 (2018)

work page 2018

[4] [3]

IEEE Trans

Chen H., He X., Weng J., Xu L.: New constructions of subspace codes using subsets of MRD codes in several blocks. IEEE Trans. Inf. Theory 66(9), 5317-5321 (2020)

work page 2020

[5] [4]

Cossidente A., Kurz S., Marino G., Pavese F.: Combining subspace codes. Adv. Math. Commun. 17(3), 536-550 (2023)

work page 2023

[6] [5]

Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory Ser. A 25(3), 226-241 (1978)

work page 1978

[7] [6]

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Etzion T., Silberstein N.: Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams. IEEE Trans. Inf. Theory 55(7), 2909-2919 (2009)

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[8] [7]

IEEE Trans

Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inf. Theory 59(2), 1004-1017 (2013)

work page 2013