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Upper bounds on the length of quasi-MDS codes

T0 review · 0 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Quasi-MDS codes cannot be longer than a partial-spread bound imported from finite geometry.

desk verdict Clean geometric reduction that legitimately tightens QMDS length bounds by importing classical partial-spread results; the necessary-not-sufficient caveat is already flagged by the authors and does not break the claims. read the letter →

arxiv 2607.26684 v1 pith:7GF3EXZX submitted 2026-07-29 cs.IT math.IT

Umberto Martínez-Peñas , Rubén Rodríguez-Ballesteros This is my paper
classification cs.ITmath.IT MSC 94B0551E2305B25
keywords QMDScodesfoldedHammingdistancesubspacepackingspartialspreadsvectorspacepartitionsfractionalMDSGriesmer-typebounds
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 how long an F_q-linear quasi-MDS (QMDS) code can be in the folded Hamming metric before its length is forced by the field size q. It translates a generator matrix into a family of subspaces whose intersection pattern encodes the minimum distance, then proves that any such family is controlled by the size of certain partial spreads. Importing the sharpest known partial-spread bounds immediately recovers the earlier Griesmer-type length bound and, in several concrete regimes, improves it by an explicit integer correction that grows with the remainder parameters. A sympathetic reader cares because QMDS codes are the nearest relatives of classical MDS codes once the dimension is not a multiple of the fold size; tighter length ceilings therefore limit how far fractional Singleton-attaining codes can exceed the classical MDS conjecture.

What carries the argument

Theorem 6, the reduction from 1-subspace packings to partial spreads: after fixing any m members of the family, the remaining (e-m)-fold intersections form a partial r_0-spread inside a subspace of dimension k-m r, forcing the binomial count of those intersections to be at most μ_q(k-m r, r_0).

What would settle it

Exhibit a QMDS code whose length exceeds the integer bound of Theorem 7, 8 or 9 for some admissible q, r, r_0, e, or prove that no such code exists by constructing a matching family of subspaces that saturates the partial-spread number.

Watch

Extended reading notes

Core claim

For a faithful family of subspaces of type [n, s, k, e] arising from a QMDS code with k = e r + r_0 (0 < r_0 < r), every integer m between 0 and e-1 satisfies the binomial inequality binom(n-m, e-m) ≤ μ_q(k-m r, r_0), where μ_q(v, t) is the maximum size of a partial t-spread in F_q^v. Substituting the Drake–Freeman, Năstase–Sissokho or vector-space-partition bounds for μ_q yields length upper bounds that recover the Ball–Lavrauw–Popatia Griesmer-type bound and strictly improve its integer form in several parameter ranges.

Load-bearing premise

The reduction only forces a local intersection condition after m members are fixed; a family can meet every local partial-spread bound and still fail the global intersection requirement that defines the code.

Editorial extensions

If this is right

  • The Ball–Lavrauw–Popatia Griesmer-type length bound is recovered simply by feeding the ordinary packing bound into the reduction with m = e-1.
  • Whenever the Năstase–Sissokho hypothesis holds, the length ceiling improves the packing-derived integer bound by exactly q^ρ-1.
  • Outside that regime the vector-space-partition correction min(q^ρ-1,(q-1)(r_0-1)) still improves the packing bound and, for intermediate ρ, can beat Drake–Freeman.
  • All improved ceilings remain asymptotic to e-1 + q^r plus a lower-order term, so the dominant growth in q is unchanged.

Reading between the lines

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

  • Because the reduction is only necessary, the true maximal length may be strictly smaller than every bound obtained from Theorem 6; the gap is already visible in the binary example of Remark 25.
  • The same reduction applies verbatim to any 1-(k,s,e)_q packing, so the length tables immediately supply new upper bounds for subspace-packing numbers A_q(k,s,1;e).
  • When r_0 divides r the exceptional packing case collapses back to the classical Griesmer expression, suggesting that the genuinely new improvements live only when the remainder is nonzero.
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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

0 major / 5 minor

Summary. The paper studies upper bounds on the length n of F_q-linear quasi-MDS (QMDS) codes of type [n,r,k,d] in the folded Hamming metric. It associates generator matrices with families of subspaces of F_q^k (kernels of the column blocks) and shows that the QMDS condition translates into an intersection condition, yielding faithful families of type [n,k-r,k,⌈k/r⌉-1]. The central technical contribution is Theorem 6: for k=er+r_0 with 0<r_0<r, every m∈{0,…,e-1} satisfies binom(n-m,e-m)≤μ_q(k-mr,r_0), where μ_q is the maximum size of a partial r_0-spread. Importing the packing bound recovers the Griesmer-type bound of Ball–Lavrauw–Popatia; importing Drake–Freeman, Năstase–Sissokho and Honold–Kiermaier–Kurz then produces the stricter length bounds of Theorems 7–9 in several regimes. The correspondence (Proposition 10, Corollary 11, Theorems 1–2) and the reduction are developed carefully, with an explicit warning (Remark 25) that the local condition is only necessary.

Significance. The work cleanly links the length problem for fractional MDS / QMDS codes to the well-developed theory of partial spreads and 1-subspace packings. The reduction of Theorem 6 is elementary linear algebra yet immediately yields concrete numerical improvements over the best previously published general bound (Ball et al.), as quantified by the integer deltas Δ_DF, Δ_NS and Δ_VSP and the accompanying tables. Because the imported geometric theorems are classical and correctly specialized, the new upper bounds are reliable and ready for use by coding theorists. The paper is therefore a solid, incremental contribution that strengthens the geometric toolkit available for folded-Hamming and additive codes.

minor comments (5)
  1. [§2.2 / §3] The exclusion of the divisible case s|k (and r|k) is stated repeatedly but never collected in one place; a single sentence in §2.2 or after Definition 19 would improve readability.
  2. [Theorem 6] In the proof of Theorem 6 the ambient space U of dimension k-mr is chosen inside the intersection of the m fixed members; it would help the reader to note explicitly that the choice of U does not affect the subsequent partial-spread bound.
  3. [§5] Tables comparing Δ_DF, Δ_NS and Δ_VSP (pp. 16–21) are useful but lack captions and are not numbered; numbering them and adding a one-line caption would make cross-reference easier.
  4. A few typographical inconsistencies appear (e.g., “QMDScodes”, missing spaces around em-dashes, “Năstase” vs “N\u{a}stase”). A light copy-edit pass would remove them.
  5. [§2.3] Reference [4] (Bartoli et al., arXiv:2509.03186) is cited for a related correspondence; a brief clarifying sentence on how the present generator-side families differ from that work would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: length bounds follow from an independent geometric reduction plus external partial-spread theorems.

full rationale

The load-bearing chain is: QMDS codes correspond (via generator blocks / kernels) to faithful subspace families of type [n,k-r,k,e] (Corollary 11, Theorem 1); Theorem 6 then proves a necessary local condition that residual (e-m)-fold intersections inside a (k-mr)-slice form a partial r0-spread, yielding binom(n-m,e-m) ≤ μ_q(k-mr,r0); packing / Drake–Freeman / Năstase–Sissokho / Honold–Kiermaier–Kurz bounds on μ_q are imported from independent finite-geometry literature and produce Theorems 7–9 and the recovered Ball et al. packing bound (Corollary 28). None of these steps defines the target length in terms of itself, fits a parameter to the claimed bound, or rests on an unverified self-citation uniqueness theorem. Self-citations to the authors’ prior QMDS paper [13] supply definitions and background only. Remark 25 explicitly records that the reduction is necessary-not-sufficient, so the derived upper bounds are not forced equalities. The derivation is self-contained against external geometric benchmarks.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

Pure finite-geometry / coding-theory paper. No empirical fits. Load-bearing background is standard linear algebra over finite fields plus three external partial-spread theorems treated as black boxes. The only paper-specific modeling choice is the generator-matrix-to-subspace-family correspondence and the decision to bound via a necessary local condition.

assumptions (6)
  • standard math Standard linear algebra and dimension formula over finite fields F_q (codimension additivity under successive intersections).
    Used throughout Definitions 7–8 and the proof of Theorem 6.
  • domain assumption Singleton-type bound for folded Hamming distance: d ≤ n − ⌈k/r⌉ + 1 (Prop. 5).
    Defines the QMDS attainment condition; taken from the authors’ prior work [13] and standard coding theory.
  • domain assumption Drake–Freeman upper bound on partial spreads μ_q(k,s) (Thm. 3 / [8]).
    Imported black-box; feeds Proposition 30 and Theorem 7.
  • domain assumption Năstase–Sissokho exact value of μ_q(k,s) when s > (q^{s0}−1)/(q−1) (Thm. 4 / [15]).
    Imported black-box; feeds Proposition 34 and Theorem 8.
  • domain assumption Honold–Kiermaier–Kurz vector-space-partition bound (Thm. 5 / [10, Cor. 7]).
    Imported black-box; feeds Proposition 38 and Theorem 9.
  • ad hoc to paper A necessary local intersection condition (fix m subspaces, remaining intersections form a partial spread) is sufficient to upper-bound global family size.
    Core modeling choice of Theorem 6; Remark 25 shows it is not always tight.
invented entities (1)
  • Faithful family of subspaces of type [n,s,k,e] independent evidence
    purpose: Geometric object equivalent (up to equivalence) to an F_q-linear code in the folded Hamming metric.
    Definition 7; standardizes the 1-(k,s,e)_q packing language for the coding application. Not a physical entity; a definitional packaging of known subspace-packing notions.

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Pith. "Pith review of Upper bounds on the length of quasi-MDS codes." pith.science (2026). https://pith.science/paper/7GF3EXZX

@misc{pith2026260726684,
  author       = {Pith},
  title        = {Pith review of: Upper bounds on the length of quasi-MDS codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GF3EXZX}},
  note         = {Machine review of arXiv:2607.26684}
}
abstract

We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, N\u{a}stase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.

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Works this paper leans on

17 extracted references · 1 linked inside Pith

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