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MSRD-constructible block Ferrers diagrams achieve the Etzion–Silberstein bound, and the problem reduces to block-triangular diagrams for many distances.

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2026-07-10 10:48 UTC pith:GQSOGFWL

load-bearing objection Clean block-analogue of MDS-constructibility that settles new arbitrary-field cases of Etzion–Silberstein for extremal distances and strictly m-monotone diagrams.

arxiv 2607.08239 v1 pith:GQSOGFWL submitted 2026-07-09 math.CO cs.ITmath.IT

On the Etzion-Silberstein conjecture for block Ferrers diagrams

Marco Calderini , Marta Messia , Alessandro Neri This is my paper
classification math.CO cs.ITmath.IT MSC 11T7194B05
keywords rank-metric codesblock Ferrers diagramsEtzion-Silberstein conjectureMSRD-constructible diagramsum-rank metricMDS-constructibleFerrers diagram codes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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The paper studies rank-metric codes whose supports are restricted to block Ferrers diagrams—Ferrers diagrams whose dots come in fixed-size square blocks. It introduces MSRD-constructibility, the block analogue of the classical MDS-constructible property: the Etzion–Silberstein upper bound equals the dimension obtained by placing maximum sum-rank distance codes on the block diagonals. Whenever this equality holds, the construction produces optimal codes over every field larger than the longest block diagonal. The authors prove that MSRD-constructibility of a block diagram is equivalent to MDS-constructibility of its ordinary contraction when the distance is compatible with the block size, and they give lifting criteria for the remaining distances. As a result, every strictly block-monotone or initially block-convex diagram is MSRD-constructible. Finally they reduce the existence question for these diagrams to the existence of optimal codes on block-triangular diagrams, and they settle the triangular case for the extreme distances 1 through m+1 and sm−1, sm over every finite field.

Core claim

For an m-block Ferrers diagram F and distance d, the pair (F,d) is MSRD-constructible when the Etzion–Silberstein quantity ν_min(F,d) equals the dimension obtained by placing MSRD codes on its block diagonals. Construction 3.5 then yields an optimal [F,ν_min(F,d),d]_q code for every field of size at least one more than the longest contributing block diagonal. When d−1 is a multiple of m this property is equivalent to MDS-constructibility of the contracted diagram; otherwise it lifts under Singleton-index and triangular-covered conditions. Consequently every strictly m-monotone or initially m-convex block diagram is MSRD-constructible, and the existence of optimal codes for all such pairs red

What carries the argument

MSRD-constructibility of a pair (F,d): the equality ν_min(F,d)=m∑ max{0,m|DB_i∩B(F)|−d+1}, which legitimises the placement of linearized Reed–Solomon MSRD codes on the block diagonals and produces an optimal Ferrers diagram code.

Load-bearing premise

The reduction that removes the field-size restriction assumes optimal codes already exist on the relevant block-triangular diagram, and those codes are currently known only for the extreme distances treated in the paper.

What would settle it

Produce an explicit m-block Ferrers diagram that is MSRD-constructible for an intermediate distance (neither ≤m+1 nor ≥sm−1) yet admits no optimal code over a small field such as F_2, or construct the missing optimal codes for all block-triangular diagrams of intermediate distance.

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Editorial analysis

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Referee Report

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Summary. The paper investigates the Etzion–Silberstein conjecture for m-block Ferrers diagrams by introducing MSRD-constructibility: pairs (F,d) for which the Etzion–Silberstein bound equals the dimension obtained by placing MSRD codes on the block diagonals (Construction 3.5). It proves that every MSRD-constructible pair yields an optimal [F,ν_min(F,d),d]_q code whenever q is large enough (Theorem 3.8). It then relates MSRD-constructibility of an m-block diagram to MDS-constructibility of its contraction, establishing equivalence when d−1 is a multiple of m (Theorem 4.15) and giving lifting criteria in the general case (Theorems 4.23 and 4.27). As a consequence, strictly m-monotone and initially m-convex diagrams are MSRD-constructible for all admissible distances (Corollary 4.28). Finally, a reduction theorem (Theorem 5.3) shows that arbitrary-field existence for MSRD-constructible pairs reduces to the block-triangular case; optimal codes on block-triangular diagrams are constructed for the extremal distances d∈{1,…,m+1}∪{sm-1,sm} (Theorem 5.9), yielding new unconditional cases of the conjecture (Corollary 5.10).

Significance. The work supplies a natural and clean sum-rank analogue of the classical diagonal MDS construction, together with precise equivalence and lifting theorems that transfer known MDS-constructible families to the block setting. The reduction to block-triangular diagrams mirrors the strategy that settled the MDS-constructible case and isolates a concrete open problem. The new arbitrary-field results for extremal distances, obtained by explicit linear-algebra constructions over arbitrary finite fields, constitute genuine progress on the Etzion–Silberstein conjecture. All central arguments rest on elementary combinatorial counting of dots after Singleton deletions and standard rank inequalities; the constructions are fully explicit.

minor comments (5)
  1. In Definition 3.2 and subsequent block notation, the authors consistently write F=[[h1,…,hs]]m; a brief parenthetical reminder that the double brackets indicate the block-column heights would help readers who jump into Section 3.
  2. Lemma 4.4 asserts that the minimum of the u j is attained only at the “aligned” indices km and km+r. The short counting argument given is correct, but a one-line reference to the fact that mixed row/column deletions inside a single block never improve the count would make the claim self-contained.
  3. In the proof of Theorem 4.23 (case δ=s), the inequality chain that forces u min(Fm,d)=m(m-r) relies on the Singleton bound for the diagonal construction; a forward reference to Theorem 2.11 would clarify the source of the upper bound.
  4. Example 4.24 is useful, yet the diagrams are described only by column vectors. Adding a short textual description of which dots survive the Singleton deletion would make the counter-example easier to verify by eye.
  5. The field-size hypothesis in Construction 3.5 (q≥ DBmax+1) is sharp for linearized Reed–Solomon codes, but a parenthetical remark that any family of MSRD codes of the required parameters would suffice would emphasize the modularity of the construction.

Circularity Check

0 steps flagged

No significant circularity: MSRD-constructibility, diagonal construction, contraction equivalences, and triangular reduction are all definitional or constructive without self-referential loops or fitted inputs.

full rationale

The paper defines MSRD-constructibility (Def. 3.7) exactly as equality of the Etzion–Silberstein quantity ν_min(F,d) with the sum of dimensions of MSRD codes placed on block diagonals; Construction 3.5 then realises that dimension by direct sum of known MSRD codes (Thm 2.28) and obtains the distance lower bound from the elementary rank inequality rk([A B; 0 D]) ≥ rk(A)+rk(D) applied recursively to block diagonals (Lemma 3.6). Theorem 3.8 is therefore the tautological statement that the construction meets the bound when the equality holds by definition. The subsequent equivalences and lifting criteria (Thms 4.15, 4.23, 4.27) are pure combinatorial identities relating ν_min of a diagram to that of its m-block expansion, proved by counting dots after row/column deletions that respect the block partition (Lemmas 4.4–4.6) together with the classical MDS-constructibility characterisation of the contraction (Lemma 4.22 from the literature). The reduction Theorem 5.3 likewise uses only the standard inclusion/puncturing lemmas (Lemma 2.21) and a block-adapted version of a known diagonal-support argument (Lemma 4.31); it does not assume the target result. Explicit matrix constructions for the extremal triangular cases (Lemmas 5.7–5.8) are given by concrete F_q-linear maps on field extensions and are verified by direct kernel-dimension computation; they do not rely on any prior claim of the paper. Self-citations to the authors’ earlier MDS work [17] supply background lemmas that are independently proved there and are used only as black-box combinatorial facts, never as uniqueness theorems that force the present conclusions. No parameter is fitted to data, no quantity is redefined in terms of itself, and every existence claim is either conditional on a stated field-size hypothesis or proved by an explicit construction. The derivation chain is therefore free of circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The paper rests on standard finite-field linear algebra, the classical Singleton-type bounds for rank and sum-rank metrics, the existence of linearized Reed–Solomon MSRD codes for q ≥ length, and previously proved MDS-constructibility of strictly monotone diagrams. No free parameters are fitted; the only invented notions are definitional (MSRD-constructible pair, block contraction).

axioms (3)
  • domain assumption Existence of [(m imes m)^t , m(tm-d+1), d]_q MSRD codes whenever t ≤ q-1 (linearized Reed–Solomon codes).
    Invoked in Construction 3.5 and Theorem 2.28; taken from Martínez-Peñas 2018.
  • standard math Etzion–Silberstein upper bound k ≤ u_min(F,d) for any Ferrers-diagram rank-metric code.
    Theorem 2.11, used throughout as the optimality benchmark.
  • domain assumption Strictly monotone (resp. initially convex) Ferrers diagrams are MDS-constructible for every distance (Neri–Stanojkovski).
    Used in Corollary 4.28 to transfer constructibility to the block setting.
invented entities (2)
  • MSRD-constructible pair (F,d) independent evidence
    purpose: Captures when the block-diagonal MSRD construction meets the Etzion–Silberstein bound.
    Definitional; independent evidence is the explicit dimension formula matching u_min.
  • m-block contraction / expansion maps φ_m, φ_m^{-1} independent evidence
    purpose: Relate block diagrams to ordinary Ferrers diagrams so that MDS results lift.
    Purely combinatorial bijection; no extra physical or algebraic postulate.

pith-pipeline@v1.1.0-grok45 · 34686 in / 2458 out tokens · 24756 ms · 2026-07-10T10:48:54.376283+00:00 · methodology

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read the original abstract

Ferrers diagram rank-metric codes are rank-metric codes with prescribed support, and their dimension is bounded from above by the Etzion--Silberstein bound. In this paper, we study this problem for block Ferrers diagrams, namely Ferrers diagrams whose dots are grouped into square blocks of a fixed size. Motivated by the diagonal construction for MDS-constructible Ferrers diagrams, we introduce the notion of MSRD-constructibility, where MDS codes on diagonals are replaced by maximum sum-rank distance (MSRD) codes on block diagonals. We show that MSRD-constructible pairs yield optimal Ferrers diagram rank-metric codes over sufficiently large finite fields. We then relate MSRD-constructibility of a block Ferrers diagram to MDS-constructibility of its contraction, proving an equivalence when the distance is compatible with the block size and giving lifting criteria in the general case. As a consequence, we obtain MSRD-constructibility for strictly block-monotone and initially block-convex diagrams. Finally, we prove a reduction to block triangular diagrams and use it to obtain new arbitrary-field cases of the Etzion--Silberstein conjecture for MSRD-constructible block Ferrers diagrams.

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