Long QMDS additive code

Alessandro Giannoni; Daniele Bartoli; Giuseppe Marino; Yue Zhou

REVIEW 3 major objections 5 minor 1 cited by

Long QMDS additive code

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Additive codes can reach lengths q^h+2 and q^h+3—beyond the classical q^h+1 barrier—while still meeting the generalized Singleton bound.

desk verdict The q^h+2 and q^h+3 additive QMDS constructions are sound; the apparent §4.2 existence gap is fillable, so the paper deserves a normal referee. read the letter →

arxiv 2509.03186 v1 pith:IDUGJJIM submitted 2025-09-03 math.CO

Daniele Bartoli , Alessandro Giannoni , Giuseppe Marino , Yue Zhou This is my paper

classification math.CO MSC 94B0551E2005B25

keywords additivecodesquasi-MDSsubspacepackingspartialspreadsdualhyperovalsVandermondedeterminantSingletonboundlong

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 asks whether codes over F_{q^h} that are linear only over the smaller field F_q—'additive' codes, whose normalized dimension r/h may be fractional—can be longer than the longest classical F_{q^h}-linear MDS codes (length q^h+1) without losing the optimal Singleton distance. It answers yes, constructing explicit 'long' additive QMDS codes of length q^h+2 for all q≥2, h≥2, and length q^h+3 whenever certain auxiliary subspaces of prescribed dimensions exist. The construction works geometrically: a set of hyperplane-like subspaces in an F_q-vector space is shown to be a (k-1)-packing—any k of them meet only at zero—which by a known correspondence yields exactly the desired code parameters. A reader should care because the result shows the classical length bound is an artifact of full F_{q^h}-linearity, and the fractional slack can be spent on extra coordinates; it also connects the extremal dually QMDS codes to dual hyperovals, a central object in finite geometry.

What carries the argument

The load-bearing identity is the Vandermonde determinant: for distinct α_1,...,α_k ∈ F_q^h, the system ∑ α_j^{i-1} x_i = 0 has only the trivial solution because det VM(α_1,...,α_k) = ∏_{i<j}(α_j - α_i) ≠ 0. The construction shapes every k-fold intersection so that it reduces to such a system—in the cases involving W_{∞1} or W_{∞2}, the special form of those subspaces first kills one or two of the field coordinates, shrinking the Vandermonde system—and the final r0 coordinates are killed by the F_q-linear independence of the basis elements 1, ξ, ..., ξ^{r0-1}. Thus the whole (k-1)-packing property is a statement about evaluating polynomials at distinct points.

What would settle it

Pick an allowed small parameter set, for instance q=2, h=6, r0=1, r1=4, r2=3, k=3. Explicitly search F_2^6 for three 4-dimensional subspaces Y1,Y2,Y3 with Y1∩Y2∩Y3={0} and three pairwise disjoint 3-dimensional subspaces S1,S2,S3; if they exist, build the code of Theorem 4.6 and compute its minimum distance by exhaustive enumeration of all 2^13 codewords. Any k-fold intersection in L other than {0}, or a minimum distance smaller than n-k+1 = 65, refutes the construction; a failed subspace search shows the q^h+3 claim has no content for that parameter set.

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Extended reading notes

Core claim

On its own terms, the central claim: the set L = {W_α} ∪ {W_{∞1}, W_{∞2}}, with W_α defined by a Vandermonde-type equation ∑ α^{i-1} x_i + α^{k-1} x_k = 0, is a (k-1)-packing—any k of its subspaces intersect only at zero. The proof reduces each case to a Vandermonde system with distinct nodes, whose determinant is nonzero, forcing the field coordinates to zero; the final r0 coordinates then vanish because 1, ξ, ..., ξ^{r0-1} are F_q-linearly independent. This packing corresponds to an additive [q^h+2, (k-1)+r0/h, q^h+3-k]^h_q QMDS code, which is longer than q^h+1. With three W_∞ subspaces and auxiliary families satisfying r1+r2 = h+r0, the length becomes q^h+3; the paper also proves the dual

Load-bearing premise

The length q^h+3 construction assumes, for every parameter choice allowed by the theorem, that three F_q-subspaces of the required dimension r1 exist with triple intersection zero, and three pairwise disjoint F_q-subspaces of dimension r2 exist as well. The paper does not prove this existence; for h=7 the dimensions cannot even be chosen (which is why h=7 is excluded), so the q^h+3 construction stands or falls with this unproved existence.

Editorial extensions

If this is right

For every prime power q, every h ≥ 2, and every k with 2 ≤ k ≤ q^h - 1, choosing any r0 with 1 ≤ r0 ≤ h/2 gives an explicit additive QMDS code of length q^h+2 and dimension (k-1)h+r0 over F_q.
When h ≥ 6, h ≠ 7 and r0 ≤ h/6, the same idea yields length q^h+3, with r1 and r2 chosen so that r1+r2 = h+r0, provided the required auxiliary subspaces exist.
For k=3 the number of 'infinite' subspaces can be increased to the size of a large partial spread, giving length q^h + g(q) with g(q) at least ∑_{i=1}^{a-1} q^{i r2+b} + 1; for k=2, length q^h + f(q) with f(q) at least ∑_{i=1}^{a-1} q^{i r0+b} + 1, and exactly (q^h-1)/(q^{r0}-1) when r0 divides h.
The duals of the long codes from Sections 4.1 and 4.2 are not QMDS, so the long codes are examples of fractional QMDS codes that are not dually QMDS.
If a faithful additive QMDS code has r = h + r0, then its dual is also QMDS; in particular, the partial-spread construction with k=2 gives dually QMDS codes of length q^h+f(q) and distance 2.

Reading between the lines

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

The paper's upper bound n ≤ k-2+q^h+(q^h-1)/(q^{r0}-1) suggests that taking r0=1 could permit lengths on the order of q^h + q^{h-1}; whether the packing constructions can be pushed to reach that regime for general k is an open question, but the k=2 and k=3 families already show that the 'long' phenomenon is not an isolated accident.
The q^h+3 construction hinges on an existence assumption for auxiliary subspaces; investigating exactly which (h, r0, r1, r2) admit such triples—and why h=7 is the only excluded small case—looks like a tractable finite-geometry problem whose answer would either complete or bound the family.
The equivalence with dual hyperovals means that classifying h-dimensional dual hyperovals would immediately settle the longest possible dually QMDS codes with fractional dimension (k-1)+1/h; since dual hyperovals are known only in even characteristic, the odd-q nonexistence is probably a shadow of a deeper parity constraint on such extremal codes.
Because the k=2 partial-spread codes are dually QMDS, they may be the more practical family for applications such as quantum stabilizer codes, where both the code and its dual need good distance; the non-dually-QMDS long codes of Section 4.1 might still be useful when only one direction matters.

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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

Quick take: the main constructions are sound, and the one apparent gap in Section 4.2 — existence of the auxiliary subspaces — is not actually a gap. You can build Y_1,Y_2,Y_3 as coordinate subspaces vanishing on complementary subsets of {1,...,h}, and pull S_1,S_2,S_3 from a Beutelspacher partial spread; the inequalities r1≤2h/3 and r2≤h/2 make it work. So both the q^h+2 and q^h+3 families stand.

What is genuinely new: the explicit Vandermonde packings give additive QMDS codes of length q^h+2 and q^h+3, beyond the usual q^h+1 ceiling for F_{q^h}-linear MDS codes. The packing viewpoint is clean, and the dual-code analysis in Section 5 gives usable conditions for the QMDS property to survive duality. The paper is also honest about provenance: Section 4.3 reproduces [1, Thm 23] in the r0|h case, and Theorem 5.10 is a new proof of [14, Thm 1]. That overlap is not concealed.

Soft spots, in proportion. First, Theorem 4.6 leans on 'same as Theorem 4.3' for cases with two W∞'s. The math works, but the proof is terser than it should be; the key point is that even when Y_1∩Y_2 is nontrivial, the Vandermonde system kills the y coordinate. Second, Theorem 5.7 has a real typo: the final case needs C = B_{q,h}, not C = B^⊥_{q,h}. The dimension computation only makes sense for B. That is minor but should be fixed. Third, the chain #Ω1 ≥ #Γ ≥ #Ω2 in Section 4.2.1 is compressed; it is recoverable by injecting (h−r1)-subspaces into the r2-spread and dualizing, but a skeptical reader will want a sentence.

I disagree with the reader's strongest concern. The §4.2 existence assumption is not load-bearing; the stress-test construction is valid, and the authors' exclusion of h=7 matches exactly where the dimension constraints cannot be satisfied. So the conditional verdict should resolve to accept the main claims.

This paper is for coding theorists and finite geometers working on additive codes or subspace packings. It deserves a serious referee. I would send it to review, asking for the Theorem 5.7 typo fix and a small expansion of the terse cases.

Referee Report

3 major / 5 minor

Summary. The paper constructs additive F_q-linear codes in F_qh^n attaining the generalized Singleton bound (QMDS) with length exceeding q^h+1. Section 3 recalls the geometric translation between additive codes and subspace packings. Section 4.1 gives a packing of q^h+2 subspaces of dimension (k-2)h+r0 in F_q^{(k-1)h+r0}, with a Vandermonde proof that any k meet trivially; Section 4.2 extends this to q^h+3 under hypotheses h≥6, h≠7 and r1+r2=h+r0; Section 4.2.1 gives a k=3 variant with q^h+g(q) blocks; Section 4.3 treats k=2 via partial spreads. Section 5 studies dual codes and shows some constructed codes are not dually QMDS; Section 6 connects dually QMDS codes to dimensional dual arcs and derives nonexistence for odd q in extremal cases.

Significance. If the gaps below are patched, the paper provides explicit long additive QMDS codes beyond the q^h+1 linear MDS bound, with elementary and verifiable arguments. The main construction is a genuine geometric contribution: it reinterprets the QMDS condition as a (k-1)-packing and uses Vandermonde systems to ensure vanishing intersections. The paper also spells out the relation between dual hyperovals and dually QMDS codes. The proofs are explicit and do not rely on fitted parameters or circular reasoning; external results are used as tools. The dual-code section is useful but contains the correctness issues noted below.

major comments (3)

[§4.2, before Theorem 4.6] The proof of Theorem 4.6 is conditional on the asserted existence of Y1,Y2,Y3 (r1-dimensional, triple intersection zero) and S1,S2,S3 (r2-dimensional, pairwise disjoint) for every allowed q,h,r0,r1,r2. This is load-bearing for the length q^h+3 construction. Existence is true and can be shown by coordinate subspaces for the Y_i (three subsets of [h] of size h-r1 with union [h]) and by Beutelspacher's bound for the S_i, but the manuscript should state this explicitly; as written it is an unstated assumption.
[§4.2.1] The step 'Since r2 ≥ h−r1, we have #Ω2 ≤ Γ' is not justified as written. If Γ is only an inclusion-maximal partial (h−r1)-spread, the inequality is false in general; a maximum spread is needed. The intended argument is to choose an (h−r1)-subspace inside each S_i; these are pairwise disjoint, so #Γ ≥ #Ω2, and then Γ^⊥ gives #Ω1 ≥ #Γ. Please include this argument and use #Γ rather than Γ.
[Theorem 5.7] The statement needs k≥3: for k=2, A_{q,h,2} is a partial-spread code and Corollary 5.9 implies its dual is QMDS, so the theorem as stated is false. Also, in the last paragraph 'let C = B^⊥_{q,h}' must be 'let C = B_{q,h}' for Theorem 5.6 to apply. Finally, the inequality dim(W∞1∩W∞2) ≥ (k−3)h+2r0 in the B_{q,h,k} case follows from dim(Y1∩Y2) ≥ 2r1−h ≥ 2r0, but this is not said; the reader is left with an unexplained jump.

minor comments (5)

[Theorem 2.2] The theorem assumes t∤r and then says the bound is tight when t divides r; state the t|r case separately to avoid confusion.
[§4.2.1] Replace 'maximal' by 'maximum' for Γ and Ω1, and define g(q)=#Ω2 after Ω2 is introduced.
[Corollary 5.9] The symbol n is used both for the length and in n−1−r0/h; use |Ω| or specify n = q^h+f(q) explicitly.
[Section 4.1] State the lower bound 1 ≤ r0 in the parameter conditions; the text only says r0 ≤ h/2.
[Theorem 5.7] The notation A^⊥_{q,h,k}, B^⊥_{q,h,k}, B^⊥_{q,h} should be accompanied by the intended parameter ranges (especially k≥3) so the statement is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constructions are explicit and rely on external theorems; no fitted inputs or self-citation chain is load-bearing.

full rationale

The central claims are constructive, not derived from their own targets. In Section 4.1, the set L is defined explicitly and Theorem 4.3 proves the (k-1)-packing property by a Vandermonde argument; Remark 4.4 then applies the independent equivalence of Theorem 3.8 between 1-packings and additive codes. The length n=q^h+2 and the distance relation n-d=k-1 are direct consequences of the construction, not fitted parameters renamed as predictions. Section 4.2 follows the same pattern. The only assumption that is not fully justified in the text is the existence of the auxiliary subspaces Y1,Y2,Y3 and S1,S2,S3. The paper simply says "Let Y1, Y2, Y3 be three F_q-subspaces..." and "S1, S2, S3 be three F_q-subspaces...". This is an omitted existence proof, but it is not circular: the existence of such subspaces does not encode the target QMDS parameters, and for the allowed parameter ranges such subspaces can be constructed (coordinate subspaces for the Y_i, and Beutelspacher's partial spread bound, Theorem 2.3, for the S_i). Section 4.2.1's use of maximal partial spreads and orthogonal complements is also geometric fact external to the code construction, not a self-citation. The upper bound used as a benchmark is [1, Theorem 4.1], by Ball--Lavrauw--Popatia, not by any of the present authors; no load-bearing self-citation occurs. Section 6 connects dual arcs to dually QMDS codes via Theorem 5.10 and external DHO results; this is application of known facts rather than circular renaming. Overall, the paper is self-contained against external bounds and contains no step where an input is equivalent by construction to the claimed output.

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

No numerically fitted parameters. The central constructions rely on existence of particular subspaces in finite geometries; these are domain assumptions rather than invented entities.

assumptions (5)

domain assumption There exist two disjoint r0-dimensional F_q-subspaces S1,S2 of F_{q^h} when r0 ≤ h/2.
Used in Section 4.1 to define W∞1 and W∞2; standard but not proved.
domain assumption For h≥6, h≠7, r0≤h/6, r1≤2h/3, r2≤h/2, r1+r2=h+r0, there exist r1-dimensional Y_i with Y1∩Y2∩Y3=0 and pairwise disjoint r2-dimensional S_i.
Load-bearing for Theorem 4.6; no proof or citation supplied.
standard math Beutelspacher lower bound for partial spreads (Theorem 2.3).
Used in Sections 4.2.1 and 4.3 to lower-bound f(q) and g(q).
domain assumption Upper bound on length of additive QMDS codes from [1] (Theorem 4.1).
Used to frame optimality; cited.
domain assumption Dimensional dual hyperovals exist only over even characteristic (Theorem 2.8).
Used for Corollaries 6.2 and 6.3.

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Pith. "Pith review of Long QMDS additive code." pith.science (2026). https://pith.science/paper/IDUGJJIM

@misc{pith2026250903186,
  author       = {Pith},
  title        = {Pith review of: Long QMDS additive code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDUGJJIM}},
  note         = {Machine review of arXiv:2509.03186}
}

abstract

We investigate additive codes, defined as $\mathbb{F}_q$-linear subspaces $C \subseteq \mathbb{F}_{q^h}^n$ of length $n$ and dimension $r$ over $\mathbb{F}_q$. An additive code is said to be of type $[n, r/h, d]_q^h$, where $d$ denotes the minimum Hamming distance and the normalized dimension $r/h$ may be fractional. A central object of interest is the class of quasi-MDS (QMDS) codes, those additive codes achieving the generalized Singleton bound: $$ d = n - \left\lceil \frac{r}{h} \right\rceil + 1. $$ In this work, we construct explicit families of additive QMDS codes whose lengths exceed those of the best-known $\mathbb{F}_{q^h}$-linear MDS codes which is $q^h+1$, and we will call these types of codes ``Long'' . By leveraging $\mathbb{F}_q$-linearity and geometric tools like partial spreads and dimensional dual arcs, we show that additive structures allow longer codes without sacrificing optimality in distance. We also examine dual codes and give conditions under which the QMDS property is preserved under duality.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Upper bounds on the length of quasi-MDS codes
cs.IT 2026-07 accept novelty 6.0 of 10

A reduction from QMDS subspace families to partial spreads yields tighter length upper bounds than the Ball–Lavrauw–Popatia Griesmer-type bound in several regimes.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

and Popatia, T

Ball, S., Lavrauw, M. and Popatia, T. Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes. Des. Codes Cryptogr. 93, 175–196 (2025). https://doi.org/10.1007/s10623-024-01519-2

work page doi:10.1007/s10623-024-01519-2 2025

[2] [2]

and Popatia, T

Ball, S., Lavrauw, M. and Popatia, T. Correction to: Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes MDS codes. Des. Codes Cryptogr. https://doi.org/10.1007/s10623-025-01681-1

work page doi:10.1007/s10623-025-01681-1

[3] [3]

Partial spreads in finite projective spaces and partial designs

Beutelspacher, A. Partial spreads in finite projective spaces and partial designs. Math Z 145, 211–229 (1975). https://doi.org/10.1007/BF01215286

work page doi:10.1007/bf01215286 1975

[4] [4]

Low-density MDS codes and their applications,

D. Boucher and F. Ulmer, “Low-density MDS codes and their applications,” Finite Fields and Their Applications , vol. 58, pp. 157–178, 2019

work page 2019

[5] [5]

On Generalized k-Arcs in PG(2n, q)

Cooperstein, B., Thas, J. On Generalized k-Arcs in PG(2n, q). Annals of Com- binatorics 5, 141–152 (2001). https://doi.org/10.1007/PL0000129

work page doi:10.1007/pl0000129 2001

[6] [6]

On d-Dimensional Dual Hyperovals

Del Fra, Alberto. “On d-Dimensional Dual Hyperovals.” Geometriae Dedicata 79 (2000): 157-178

work page 2000

[7] [7]

Dempwolff, U. (2022). Dimensional Dual Hyperovals—An Updated Survey. In: Ashraf, M., Ali, A., De Filippis, V. (eds) Algebra and Related Topics with Appli- cations. ICARTA 2019. Springer Proceedings in Mathematics and Statistics, vol

work page 2022

[8] [8]

Near-MDS codes,

S. M. Dodunekov and I. N. Landgev, “Near-MDS codes,” Journal of Geometry , vol. 54, no. 1–2, pp. 30–43, 1995

work page 1995